research article on some approximation theorems for power...
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Research ArticleOn Some Approximation Theorems for Power 119902-BoundedOperators on Locally Convex Vector Spaces
Ludovic Dan Lemle
Department of Electrical Engineering and Industrial Informatics Politehnica University of Timisoara 331128 Hunedoara Romania
Correspondence should be addressed to Ludovic Dan Lemle danlemlefihuptro
Received 21 May 2014 Accepted 28 July 2014 Published 18 August 2014
Academic Editor Antonio M Peralta
Copyright copy 2014 Ludovic Dan LemleThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper deals with the study of some operator inequalities involving the power 119902-bounded operators along with the most knownproperties and results in the more general framework of locally convex vector spaces
1 Introduction
Let 119883 be a Hausdorff locally convex vector space over thecomplex field C By calibration for the locally convex space119883 we understand a family P of seminorms generating thetopology 120591P of 119883 in the sense that this topology is thecoarsest with respect to the fact that all the seminorms inPare continuous Such a family of seminorms was used by theauthor andWu [1] and many others in different contexts (see[2ndash5])
It is well known that calibrationP is characterized by theproperty that the set
119864 (119901 120598) = 119909 isin 119883 119901 (119909) lt 120598 120598 gt 0 119901 isin P (1)
is a neighborhood subbase at 0 Denote by (119883P) the locallyconvex space119883 endowed with calibrationP
Recall that a locally convex algebra is an algebra witha locally convex topology in which the multiplication isseparately continuous Such an algebra is said to be locally119898-convex (lmc) if it has a neighborhood base U at 0 suchthat each 119880 isin U is convex and balanced (ie 120582119880 sube 119880 for|120582| le 1) and satisfies the property 1198802 sube 119880
Any algebra with identity will be called unital It iswell known that unital locally 119898-convex algebra A ischaracterized by the existence of calibrationP such that each
119901 isin P is submultiplicative (ie 119901(119909119910) le 119901(119909) 119901(119910) for all119909 119910 isin A) and satisfies 119901(119890) = 1 where 119890 is the unit element
An element 119886 of locally convex algebra A is said to bebounded inA if there exists 120572 isin C such that the set (120572119909)119899
119899ge1
is bounded inA (see [6]) The set of all bounded elements inA will be denoted byA
0
Let Cinfin
= C cup infin be the Alexandroff one-pointcompactification of C Following Waelbroeck [7 8] weintroduce the following
Definition 1 We call resolvent set in the Waelbroeck sense ofan element 119909 from a locally convex unital algebra (119883P) theset of all elements 120582
0isin Cinfin
for which there exists 119881 isin V1205820
such that the following conditions hold
(a) the element 120582119890 minus 119909 is invertible in 119883 for any 120582 isin 119881 infin
(b) the set (120582119890minus119909)minus1 120582 isin 119881infin is bounded in (119883P)
The resolvent set inWaelbroeck sense of an element 119909willbe denoted by 120588
119882(119909) The Waelbroeck spectrum of 119909 will be
defined as
120590119882(119909) = C
infin 120588119882(119909) (2)
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 513162 5 pageshttpdxdoiorg1011552014513162
2 The Scientific World Journal
2 119902-Bounded Operators
Following Michael [9] (see also [2 10]) we introduce thefollowing
Definition 2 We say that a linear operator 119879 119883 rarr 119883 is119902-bounded (quotient-bounded) with respect to P if for any119901 isin P there exists 119888
119901gt 0 such that
119901 (119879119909) le 119888119901119901 (119909) forall119909 isin 119883 (3)
Denote by119876P(119883) the set which consists of all 119902-boundedoperators with respect to calibrationP
For a seminorm 119901 isin P the application 119901 119876P(119883) rarr R
defined as
119901 (119879) = inf 119903 gt 0 119901 (119879119909) le 119903119901 (119909) forall119909 isin 119883 (4)
is also a seminorm Note that
119901 (11987911198792) le 119901 (119879
1) 119901 (119879
2) 119879
1 1198792isin 119876P (119883) 119901 isin P (5)
We denote by P the family of seminorms 119901 119901 isin P Thespace 119876P(119883) will be endowed with a topology 120591P generatedby P Remark that [9 Proposition 24(j)] implies that underthis topology 119876P(119883) becomes a Hausdorff locally 119898-convextopological algebra (in the sense of [9 Definition 21])
If 119879 isin 119876P(119883) the P-spectral radius denoted by 119903P(119879)is considered as the boundedness radius in the sense of Allan[6] (see also [11ndash13])
119903P (119879) = inf 120582 gt 0 the sequence ((120582minus1119879)119899
)119899isinN
is bounded in 119876P (119883)
(6)
where by common consent inf 0 = +infinThe set of all bounded elements in119876P(119883)will be denoted
by (119876P(119883))0 (see [12]) It easily follows from [6 Proposition214(ii)] that
(119876P (119883))0 = 119879 isin 119876P (119883) 119903P (119879) lt infin (7)
For 119879 isin (119876P(119883))0 we denote by 120588119882(119879) the Waelbroeckresolvent set of 119879 and by 120590
119882(119879) the Waelbroeck spectrum of
119879 The function
120588119882(119879) ni 120582 997891997888rarr 119877 (120582 119879) = (120582119868 minus 119879)
minus1isin (119876P (119883))0
(8)
is called the resolvent function of 119879 It is well known that
119877 (120582 119879) =
infin
sum
119899=0
119879119899
120582119899+1 (9)
In this paper we evaluate the behaviour of the power of a119902-bounded operator from the algebra (119876P(119883))0 by some typeof approximationsThemain results have been announced in[14]
3 The Main Results
We continue to employ the notations from the previoussections and we will introduce two types of operatorialapproximations for operators from the algebra (119876P(119883))0
which approximate a given operator 119879 on a convergentpower bounded series The power boundedness problem foroperators acting on Banach spaces was largely developed invarious frameworks by many authors (see [15ndash17])
In the following using the functional calculus from the(119876P(119883))0 algebra (see [7 8]) some important boundednessproperties are obtained Denote Nlowast = N 0 First we havethe following
Theorem 3 If 119879 isin (119876P(119883))0 satisfies
sup119901isinP
119901 (119879119896) le 119862 (10)
for 119896 isin Nlowast then
sup119901isinP
119901 [119877(120582 119879)119896] le
119862
(|120582| minus 1)119896 (11)
for 119896 isin Nlowast and for all 120582 isin C with |120582| gt 1
Proof Assume that sup119901isinP119901(119879
119896) le 119862 for 119896 isin Nlowast Since
119877 (120582 119879) =
infin
sum
119895=0
119879119895
120582119895+1 (12)
for |120582| gt 1 then by using the generalized binomial formulawe get
119877(120582 119879)119896= 120582minus119896(119868 minus
119879
120582)
minus119896
=1
120582119896
infin
sum
119895=0
(119895 + 119896 minus 1
119895)119879119895
120582119895 (13)
from where we deduce
119901 [119877(120582 119879)119896] le
119862
|120582|119896sdot
infin
sum
119895=0
(119895 + 119896 minus 1
119895)(
1
|120582|)
119895
=119862
|120582|119896sdot
1
(1 minus 1 |120582|)119896=
119862
(|120582| minus 1)119896
(14)
for any 119896 isin Nlowast and any 119901 isin P Therefore the conclusion isverified
Conversely we have the following
Theorem 4 If 119879 isin (119876P(119883))0 and
sup119901isinP
119901 [119877 (120582 119879)] le119862
|120582| minus 1 (15)
for all 120582 isin C with |120582| gt 1 then
sup119901isinP
119901 (119879119896) le 119862119890 (119896 + 1) (16)
for 119896 isin Nlowast
The Scientific World Journal 3
Proof Let us suppose condition 119901[119877(120582 119879)] le 119862(|120582| minus 1) istrue for all 119901 isin P for any 119896 isin Nlowast and |120582| gt 1 For 119896 isin Nlowast
fixed by choosing the integration path Γ |120582| = 1 + 1119896 withthe aid of the functional calculus from the algebra (119876P(119883))0we obtain
119879119896=1
2120587119894intΓ
120582119896119877 (120582 119879) 119889120582 (lowast)
Thus for all 119901 isin P we have
119901 (119879119896) le
1
2120587intΓ
|120582|119896119901 (119877 (120582 119879)) 119889120582
le1
2120587sdotmax120582isinΓ
|120582|119896sdotmax120582isinΓ
119862
|120582| minus 1sdot intΓ
119889120582
le1
2120587sdot (1 +
1
119896)
119896
sdot 119862119896 sdot 2120587 (1 +1
119896) le 119862119890 (119896 + 1)
(17)
which implies the desired resultMoreover we can formulate the following
Theorem 5 If 119879 isin (119876P(119883))0 and
sup119901isinP
119901 [119877(120582 119879)119896] le
119862
(|120582| minus 1)119896 (18)
for 119896 isin Nlowast and for all 120582 isin C with |120582| gt 1 then
sup119901isinP
119901 (119879119896) le 119862
119896119890119896
119896119896le 119862radic2120587 (119896 + 1) 119896 isin N
lowast (19)
Proof Integrating (lowast) by parts 119895minus1 times for 119895 gt 2 we obtain
119879119896=(minus1)119895minus1
2120587119894intΓ
(119895 + 1)120582119896+119895minus1
(119896 + 1) sdot sdot sdot (119896 + 119895 minus 1)119877(120582 119879)
119895119889120582 (20)
Now choosing Γ the circle of radius 1 + 119895119896 and by usingthe hypothesis for 119895 rarr infin we get
sup119901isinP
119901 (119879119896) le 119862
119896119890119896
119896119896le 119862radic2120587 (119896 + 1) (21)
The last inequality was obtained by using Stirlingrsquos approxi-mation
Now for 119879 isin (119876P(119883))0 we introduce (see [18]) thefollowing
Definition 6 The Yosida approximation 119884(120582 119879) of 119879 for 120582 isin120588119882(119879) cap C is defined as
119884 (120582 119879) = 120582119879119877 (120582 119879) (22)
Next theorem shows how an operator from the (119876P(119883))0
algebra is related to its Yosida approximation
Theorem 7 The Yosida approximation 119884(120582 119879) is analytic for120582 isin 120588119882(119879) cap C and the series representation
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895(23)
converges for |120582| gt 119903P(119879) Moreover
(1) 119884(120582 119879) = 1205822119877(120582 119879) minus 120582119868(2) if there exists 119901 isin P such that 119903P(119879) lt 119901(119879) then
119901 (119884 (120582 119879) minus 119879) le119901 (1198792)
|120582| minus 119901 (119879) (24)
for |120582| gt 119901(119879)(3) 120590119882(119884(120582 119879)) = 119911(1 minus 119911120582) 119911 isin 120590
119882(119879)
Proof By evaluating119884(120582 119879) in terms of the resolvent119877(120582 119879)for |120582| gt 119903P(119879) we obtain
119884 (120582 119879) = 120582119879119877 (120582 119879) = 120582119879(120582119868 minus 119879)minus1
= 120582119879 sdot
infin
sum
119895=0
119879119895
120582119895+1=
infin
sum
119895=0
119879119895+1
120582119895
(25)
fromwhere it follows that the assertion of the theorem is trueMoreover
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895= 120582119868 + 119879 +
1198792
120582+ sdot sdot sdot +
119879119899+1
120582119899+ sdot sdot sdot minus 120582119868
= 1205822
infin
sum
119895=0
119879119895
120582119895+1minus 120582119868 = 120582
2119877 (120582 119879) minus 120582119868
(26)
so (1) is trueTo prove (2) one can observe that from
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895 (27)
it follows that
119884 (120582 119879) minus 119879 =
infin
sum
119895=0
1198792
120582(119879119895
120582119895) (28)
on a set for which |120582| gt 119903P(119879) Moreover
119901 (119884 (120582 119879) minus 119879) le
infin
sum
119895=0
119901(1198792
120582)119901(
119879119895
120582119895)
le 119901(1198792
120582) sdot
infin
sum
119895=0
119901(119879
120582)
119895
= 119901(1198792
120582)
1
1 minus 119901 (119879120582)=
119901 (1198792)
|120582| minus 119901 (119879)
(29)
for |120582| gt 119901(119879) gt 119903P(119879)A simple reasoning shows that 119877(120582 119879) isin (119876P(119883))0 then
it follows 119884(120582 119879) isin (119876P(119883))0From [19 Theorem 3114] for |120582| gt |119911| we have
120590119882(119884 (120582 119879)) = 119884 (120582 120590
119882(119879)) (30)
for all 119911 isin 120590119882(119879) and
119884 (120582 119911) =
infin
sum
119895=0
119911119895+1
120582119895(31)
on |120582| gt |119911| which could be written as 119884(120582 119911) = 119911(1 minus 119911120582)for any 119911 isin 120590
119882(119879) so (3) is proved
4 The Scientific World Journal
Below we state an equivalence between a power boundedoperator from the (119876P(119883))0 algebra and the power of itsYosida approximation
Theorem 8 Let119879 isin (119876P(119883))0 and119884(120582 119879) its Yosida approx-imation Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119888 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119884(120582 119879)
119896) le 119888(1 minus 1|120582|)
119896 for any 119896 isin Nlowast
and for all 120582 isin C with |120582| gt 1
Proof Property (i) implies 119903P(119879) le 1 so that the argumenta-tion given in the proof of Theorem 7 implies that any 120582 isin C
with |120582| gt 1 belongs to the resolvent set of 119879 Hence usingthe generalized binomial formula we get
119884(120582 119879)119896=
infin
sum
119895=0
(119896 + 119895 minus 1
119895)119879119895+119896
120582119895 (32)
Now by applying (i) again we obtain
119901 (119884(120582 119879)119896) le 119888
infin
sum
119895=0
(119896 + 119895 minus 1
119895)(
1
|120582|)
119895
=119888
(1 minus 1 |120582|)119896
(33)
for any 119901 isin P whence by passing to supremum theinequality (ii) holds
Conversely (i) is a direct consequence of (ii)
For 120583 isin 120588119882(119879) consider now the following Mobius
transformation (see [20])
120595120582(120583) =
(120582 minus 1) 120583
120582 minus 120583 if 120582 = infin
120583 if 120582 = infin(34)
Definition 9 TheMobius approximation of 119879 is defined as
119860 (120582 119879) = 120595120582(119879) (35)
Proposition 10 119860(120582 119879) is holomorphic in 120582 isin 120588119882(119879) cap C
and satisfies
119860 (120582 119879) = (1 minus1
120582)119884 (120582 119879) 120582 = 0 (36)
Proof Let 120582 isin 120588119882(119879) cap C 0 By evaluating the right
member of the above equality we get successively
(1 minus1
120582)119884 (120582 119879)
= (1 minus1
120582) 120582119879119877 (120582 119879) = (120582 minus 1) 119879119877 (120582 119879)
=(120582 minus 1) 119879
120582119868 minus 119879= 119860 (120582 119879)
(37)
for 120582 = infin If 120582 = infin then from Definition 9 we have119860(120582 119879) = 119879 On the other side (1 minus 1120582)119884(120582 119879) convergesto 119879 when 120582 rarr infin
A similar result as in Theorem 8 is given below
Theorem 11 Let 119879 isin (119876P(119883))0 and 119860(120582 119879) its approxima-tion as above Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119862 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119860(120582 119879)
119896) le 119862 for any 119896 isin Nlowast and for every
120582 isin C with |120582| gt 1
Proof FromTheorem 8 for 119879 isin (119876P(119883))0
sup119901isinP
119901 (119879119896) le 119862 (38)
is equivalent to
sup119901isinP
119901 (119884(120582 119879)119896) le
119862
(1 minus 1 |120582|)119896 (39)
The conclusion follows taking into account that
119860(120582 119879)119896= (1 minus
1
120582)
119896
sdot 119884(120582 119879)119896 (40)
for 119896 isin Nlowast
4 Application
For 119871 gt 0 let 119883 = C[0 119871] be the space of continuous func-tions on [0 119871] endowed with the norm |119906|
119871= max
[0119871]|119906(119905)|
Consider 119879 119883 rarr 119883 given by
119879119906 (119905) = int
119905
0
119906 (119904) 119889119904 (41)
Following [19] we see that the resolvent of 119879 is given by
119877 (120582 119879) 119906 (119905) =1
120582119906 (119905) +
1
1205822int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (42)
the Yosida approximation of 119879 is
119884 (120582 119879) 119906 (119905) = int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (43)
and the Mobius approximation of 119879 is
119860 (120582 119879) 119906 (119905) = (1 minus1
120582)int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (44)
Remark that for all 119906 isin C[0 119871] we have
|119879119906|119871 = max119905isin[0119871]
|119879119906 (119905)|
le max119905isin[0119871]
int
119905
0
|119906 (119904)| 119889119904
le max119905isin[0119871]
|119906 (119905)| int
119871
0
119889119904 = |119906|119871 sdot 119871
(45)
The above implies that 119879 is a contraction for 119871 le 1
The Scientific World Journal 5
If 119871 gt 1 then we can introduce for each 120576 gt 0 thefollowing norm onC[0 119871]
119906120576 = max119905isin[0119871]
119890119905120576|119906 (119905)| 119906 isin C [0 119871] (46)
Then a simple computation gives that
119879119906120576lt 120576119906120576 119906 isin C [0 119871] (47)
On the other hand
119906120576le |119906|119871 le 119890
119871120576119906120576 (48)
Remark that by Theorem 11 for all 120582 gt 1 we get
|119860 (120582 119879)|119871 = (120582 minus 1) (119890119879120582minus 1) le 1 (49)
if and only if |119879|119871le 1
It is clear that for estimating the powers of 119879 it seemsto be better to use the Yosida approximation or Mobiusapproximation than the resolvent approximation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the anonymous referees for theirvery careful reading and for useful suggestions that helpedin better exposing this material
References
[1] L D Lemle and L M Wu ldquoUniqueness of C0-semigroups
on a general locally convex vector space and an applicationrdquoSemigroup Forum vol 82 no 3 pp 485ndash496 2011
[2] R T Moore ldquoBanach algebras of operators on locally convexspacesrdquo Bulletin of the American Mathematical Society vol 75pp 68ndash73 1969
[3] F Pater ldquoProperties of multipliers on special algebras withapplication to signal processingrdquo Acta Technica NapocensisApplied Mathematics and Mechanics vol 54 pp 313ndash318 2011
[4] F Pater ldquoA multiplier algebra representation with applica-tion to harmonic signal modelsrdquo in Proceedings of the AIPInternational Conference on Numerical Analysis and AppliedMathematics vol 1479 pp 1075ndash1078 Kos Greece September2012
[5] M Kostic ldquoAbstract Volterra equations in locally convexspacesrdquo Science China Mathematics vol 55 no 9 pp 1797ndash1825 2012
[6] G R Allan ldquoA spectral theory for locally convex alebrasrdquoProceedings of the LondonMathematical Society vol 15 pp 399ndash421 1965
[7] L Waelbroeck Etude Spectrale des Algebres Completes vol 31Academie Royale de Belgique Classe des sciences MemoiresCol 1960
[8] L Waelbroeck ldquoAlgebres commutatives elements reguliersrdquoBulletin of the Belgian Mathematical Society vol 9 pp 42ndash491957
[9] E A Michael ldquoLocally multiplicatively-convex topologicalalgebrasrdquo Memoirs of the American Mathematical Society vol1952 no 11 79 pages 1952
[10] G A Joseph ldquoBoundedness and completeness in locally convexspaces and algebrasrdquo Journal of the Australian MathematicalSociety vol 24 no 1 pp 50ndash63 1977
[11] F G Bonales and R V Mendoza ldquoExtending the formula tocalculate the spectral radius of an operatorrdquo Proceedings of theAmerican Mathematical Society vol 126 no 1 pp 97ndash103 1998
[12] F Pater and T Binzar ldquoOn some ergodic theorems for a uni-versally bounded operatorrdquoCarpathian Journal of Mathematicsvol 26 no 1 pp 97ndash102 2010
[13] F Pater and L D Lemle ldquoOn some multiplication operatoralgebra problem with application to stochastic signal modelsrdquoin Proceedings of the International Conference on NumericalAnalysis and Applied Mathematics vol 1558 of AIP ConferenceProceedings pp 1661ndash1664 Rhodes Greece September 2013
[14] F Pater L D Lemle and T Binzar ldquoOn some Yosida typeapproximation theoremsrdquo in Proceedings of the InternationalConference on Numerical Analysis and Applied Mathematicsvol 1168 ofAIP Conference Proceedings pp 521ndash524 RethymnoCrete September 2009
[15] Y Katznelson and L Tzafriri ldquoOn power bounded operatorsrdquoJournal of Functional Analysis vol 68 no 3 pp 313ndash328 1986
[16] B Nagy and J Zemanek ldquoA resolvent condition implying powerboundednessrdquo Studia Mathematica vol 134 no 2 pp 143ndash1511999
[17] O Nevanlinna ldquoOn the growth of the resolvent operator forpower bounded operators Linear Operatorsrdquo Banach CenterPublications vol 38 pp 247ndash264 1997
[18] K Yosida ldquoOn the differentiability and the representation ofone-parameter semi-group of linear operatorsrdquo Journal of theMathematical Society of Japan vol 1 pp 15ndash21 1948
[19] O Nevanlinna Convergence of Iterations for Linear EquationsBirkhauser Basel Switzerland 1993
[20] A L Shields ldquoOn Mobius bounded operatorsrdquo Acta Universi-tatis Szegediensis Acta Scientiarum Mathematicarum vol 40no 3-4 pp 371ndash374 1978
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2 The Scientific World Journal
2 119902-Bounded Operators
Following Michael [9] (see also [2 10]) we introduce thefollowing
Definition 2 We say that a linear operator 119879 119883 rarr 119883 is119902-bounded (quotient-bounded) with respect to P if for any119901 isin P there exists 119888
119901gt 0 such that
119901 (119879119909) le 119888119901119901 (119909) forall119909 isin 119883 (3)
Denote by119876P(119883) the set which consists of all 119902-boundedoperators with respect to calibrationP
For a seminorm 119901 isin P the application 119901 119876P(119883) rarr R
defined as
119901 (119879) = inf 119903 gt 0 119901 (119879119909) le 119903119901 (119909) forall119909 isin 119883 (4)
is also a seminorm Note that
119901 (11987911198792) le 119901 (119879
1) 119901 (119879
2) 119879
1 1198792isin 119876P (119883) 119901 isin P (5)
We denote by P the family of seminorms 119901 119901 isin P Thespace 119876P(119883) will be endowed with a topology 120591P generatedby P Remark that [9 Proposition 24(j)] implies that underthis topology 119876P(119883) becomes a Hausdorff locally 119898-convextopological algebra (in the sense of [9 Definition 21])
If 119879 isin 119876P(119883) the P-spectral radius denoted by 119903P(119879)is considered as the boundedness radius in the sense of Allan[6] (see also [11ndash13])
119903P (119879) = inf 120582 gt 0 the sequence ((120582minus1119879)119899
)119899isinN
is bounded in 119876P (119883)
(6)
where by common consent inf 0 = +infinThe set of all bounded elements in119876P(119883)will be denoted
by (119876P(119883))0 (see [12]) It easily follows from [6 Proposition214(ii)] that
(119876P (119883))0 = 119879 isin 119876P (119883) 119903P (119879) lt infin (7)
For 119879 isin (119876P(119883))0 we denote by 120588119882(119879) the Waelbroeckresolvent set of 119879 and by 120590
119882(119879) the Waelbroeck spectrum of
119879 The function
120588119882(119879) ni 120582 997891997888rarr 119877 (120582 119879) = (120582119868 minus 119879)
minus1isin (119876P (119883))0
(8)
is called the resolvent function of 119879 It is well known that
119877 (120582 119879) =
infin
sum
119899=0
119879119899
120582119899+1 (9)
In this paper we evaluate the behaviour of the power of a119902-bounded operator from the algebra (119876P(119883))0 by some typeof approximationsThemain results have been announced in[14]
3 The Main Results
We continue to employ the notations from the previoussections and we will introduce two types of operatorialapproximations for operators from the algebra (119876P(119883))0
which approximate a given operator 119879 on a convergentpower bounded series The power boundedness problem foroperators acting on Banach spaces was largely developed invarious frameworks by many authors (see [15ndash17])
In the following using the functional calculus from the(119876P(119883))0 algebra (see [7 8]) some important boundednessproperties are obtained Denote Nlowast = N 0 First we havethe following
Theorem 3 If 119879 isin (119876P(119883))0 satisfies
sup119901isinP
119901 (119879119896) le 119862 (10)
for 119896 isin Nlowast then
sup119901isinP
119901 [119877(120582 119879)119896] le
119862
(|120582| minus 1)119896 (11)
for 119896 isin Nlowast and for all 120582 isin C with |120582| gt 1
Proof Assume that sup119901isinP119901(119879
119896) le 119862 for 119896 isin Nlowast Since
119877 (120582 119879) =
infin
sum
119895=0
119879119895
120582119895+1 (12)
for |120582| gt 1 then by using the generalized binomial formulawe get
119877(120582 119879)119896= 120582minus119896(119868 minus
119879
120582)
minus119896
=1
120582119896
infin
sum
119895=0
(119895 + 119896 minus 1
119895)119879119895
120582119895 (13)
from where we deduce
119901 [119877(120582 119879)119896] le
119862
|120582|119896sdot
infin
sum
119895=0
(119895 + 119896 minus 1
119895)(
1
|120582|)
119895
=119862
|120582|119896sdot
1
(1 minus 1 |120582|)119896=
119862
(|120582| minus 1)119896
(14)
for any 119896 isin Nlowast and any 119901 isin P Therefore the conclusion isverified
Conversely we have the following
Theorem 4 If 119879 isin (119876P(119883))0 and
sup119901isinP
119901 [119877 (120582 119879)] le119862
|120582| minus 1 (15)
for all 120582 isin C with |120582| gt 1 then
sup119901isinP
119901 (119879119896) le 119862119890 (119896 + 1) (16)
for 119896 isin Nlowast
The Scientific World Journal 3
Proof Let us suppose condition 119901[119877(120582 119879)] le 119862(|120582| minus 1) istrue for all 119901 isin P for any 119896 isin Nlowast and |120582| gt 1 For 119896 isin Nlowast
fixed by choosing the integration path Γ |120582| = 1 + 1119896 withthe aid of the functional calculus from the algebra (119876P(119883))0we obtain
119879119896=1
2120587119894intΓ
120582119896119877 (120582 119879) 119889120582 (lowast)
Thus for all 119901 isin P we have
119901 (119879119896) le
1
2120587intΓ
|120582|119896119901 (119877 (120582 119879)) 119889120582
le1
2120587sdotmax120582isinΓ
|120582|119896sdotmax120582isinΓ
119862
|120582| minus 1sdot intΓ
119889120582
le1
2120587sdot (1 +
1
119896)
119896
sdot 119862119896 sdot 2120587 (1 +1
119896) le 119862119890 (119896 + 1)
(17)
which implies the desired resultMoreover we can formulate the following
Theorem 5 If 119879 isin (119876P(119883))0 and
sup119901isinP
119901 [119877(120582 119879)119896] le
119862
(|120582| minus 1)119896 (18)
for 119896 isin Nlowast and for all 120582 isin C with |120582| gt 1 then
sup119901isinP
119901 (119879119896) le 119862
119896119890119896
119896119896le 119862radic2120587 (119896 + 1) 119896 isin N
lowast (19)
Proof Integrating (lowast) by parts 119895minus1 times for 119895 gt 2 we obtain
119879119896=(minus1)119895minus1
2120587119894intΓ
(119895 + 1)120582119896+119895minus1
(119896 + 1) sdot sdot sdot (119896 + 119895 minus 1)119877(120582 119879)
119895119889120582 (20)
Now choosing Γ the circle of radius 1 + 119895119896 and by usingthe hypothesis for 119895 rarr infin we get
sup119901isinP
119901 (119879119896) le 119862
119896119890119896
119896119896le 119862radic2120587 (119896 + 1) (21)
The last inequality was obtained by using Stirlingrsquos approxi-mation
Now for 119879 isin (119876P(119883))0 we introduce (see [18]) thefollowing
Definition 6 The Yosida approximation 119884(120582 119879) of 119879 for 120582 isin120588119882(119879) cap C is defined as
119884 (120582 119879) = 120582119879119877 (120582 119879) (22)
Next theorem shows how an operator from the (119876P(119883))0
algebra is related to its Yosida approximation
Theorem 7 The Yosida approximation 119884(120582 119879) is analytic for120582 isin 120588119882(119879) cap C and the series representation
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895(23)
converges for |120582| gt 119903P(119879) Moreover
(1) 119884(120582 119879) = 1205822119877(120582 119879) minus 120582119868(2) if there exists 119901 isin P such that 119903P(119879) lt 119901(119879) then
119901 (119884 (120582 119879) minus 119879) le119901 (1198792)
|120582| minus 119901 (119879) (24)
for |120582| gt 119901(119879)(3) 120590119882(119884(120582 119879)) = 119911(1 minus 119911120582) 119911 isin 120590
119882(119879)
Proof By evaluating119884(120582 119879) in terms of the resolvent119877(120582 119879)for |120582| gt 119903P(119879) we obtain
119884 (120582 119879) = 120582119879119877 (120582 119879) = 120582119879(120582119868 minus 119879)minus1
= 120582119879 sdot
infin
sum
119895=0
119879119895
120582119895+1=
infin
sum
119895=0
119879119895+1
120582119895
(25)
fromwhere it follows that the assertion of the theorem is trueMoreover
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895= 120582119868 + 119879 +
1198792
120582+ sdot sdot sdot +
119879119899+1
120582119899+ sdot sdot sdot minus 120582119868
= 1205822
infin
sum
119895=0
119879119895
120582119895+1minus 120582119868 = 120582
2119877 (120582 119879) minus 120582119868
(26)
so (1) is trueTo prove (2) one can observe that from
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895 (27)
it follows that
119884 (120582 119879) minus 119879 =
infin
sum
119895=0
1198792
120582(119879119895
120582119895) (28)
on a set for which |120582| gt 119903P(119879) Moreover
119901 (119884 (120582 119879) minus 119879) le
infin
sum
119895=0
119901(1198792
120582)119901(
119879119895
120582119895)
le 119901(1198792
120582) sdot
infin
sum
119895=0
119901(119879
120582)
119895
= 119901(1198792
120582)
1
1 minus 119901 (119879120582)=
119901 (1198792)
|120582| minus 119901 (119879)
(29)
for |120582| gt 119901(119879) gt 119903P(119879)A simple reasoning shows that 119877(120582 119879) isin (119876P(119883))0 then
it follows 119884(120582 119879) isin (119876P(119883))0From [19 Theorem 3114] for |120582| gt |119911| we have
120590119882(119884 (120582 119879)) = 119884 (120582 120590
119882(119879)) (30)
for all 119911 isin 120590119882(119879) and
119884 (120582 119911) =
infin
sum
119895=0
119911119895+1
120582119895(31)
on |120582| gt |119911| which could be written as 119884(120582 119911) = 119911(1 minus 119911120582)for any 119911 isin 120590
119882(119879) so (3) is proved
4 The Scientific World Journal
Below we state an equivalence between a power boundedoperator from the (119876P(119883))0 algebra and the power of itsYosida approximation
Theorem 8 Let119879 isin (119876P(119883))0 and119884(120582 119879) its Yosida approx-imation Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119888 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119884(120582 119879)
119896) le 119888(1 minus 1|120582|)
119896 for any 119896 isin Nlowast
and for all 120582 isin C with |120582| gt 1
Proof Property (i) implies 119903P(119879) le 1 so that the argumenta-tion given in the proof of Theorem 7 implies that any 120582 isin C
with |120582| gt 1 belongs to the resolvent set of 119879 Hence usingthe generalized binomial formula we get
119884(120582 119879)119896=
infin
sum
119895=0
(119896 + 119895 minus 1
119895)119879119895+119896
120582119895 (32)
Now by applying (i) again we obtain
119901 (119884(120582 119879)119896) le 119888
infin
sum
119895=0
(119896 + 119895 minus 1
119895)(
1
|120582|)
119895
=119888
(1 minus 1 |120582|)119896
(33)
for any 119901 isin P whence by passing to supremum theinequality (ii) holds
Conversely (i) is a direct consequence of (ii)
For 120583 isin 120588119882(119879) consider now the following Mobius
transformation (see [20])
120595120582(120583) =
(120582 minus 1) 120583
120582 minus 120583 if 120582 = infin
120583 if 120582 = infin(34)
Definition 9 TheMobius approximation of 119879 is defined as
119860 (120582 119879) = 120595120582(119879) (35)
Proposition 10 119860(120582 119879) is holomorphic in 120582 isin 120588119882(119879) cap C
and satisfies
119860 (120582 119879) = (1 minus1
120582)119884 (120582 119879) 120582 = 0 (36)
Proof Let 120582 isin 120588119882(119879) cap C 0 By evaluating the right
member of the above equality we get successively
(1 minus1
120582)119884 (120582 119879)
= (1 minus1
120582) 120582119879119877 (120582 119879) = (120582 minus 1) 119879119877 (120582 119879)
=(120582 minus 1) 119879
120582119868 minus 119879= 119860 (120582 119879)
(37)
for 120582 = infin If 120582 = infin then from Definition 9 we have119860(120582 119879) = 119879 On the other side (1 minus 1120582)119884(120582 119879) convergesto 119879 when 120582 rarr infin
A similar result as in Theorem 8 is given below
Theorem 11 Let 119879 isin (119876P(119883))0 and 119860(120582 119879) its approxima-tion as above Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119862 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119860(120582 119879)
119896) le 119862 for any 119896 isin Nlowast and for every
120582 isin C with |120582| gt 1
Proof FromTheorem 8 for 119879 isin (119876P(119883))0
sup119901isinP
119901 (119879119896) le 119862 (38)
is equivalent to
sup119901isinP
119901 (119884(120582 119879)119896) le
119862
(1 minus 1 |120582|)119896 (39)
The conclusion follows taking into account that
119860(120582 119879)119896= (1 minus
1
120582)
119896
sdot 119884(120582 119879)119896 (40)
for 119896 isin Nlowast
4 Application
For 119871 gt 0 let 119883 = C[0 119871] be the space of continuous func-tions on [0 119871] endowed with the norm |119906|
119871= max
[0119871]|119906(119905)|
Consider 119879 119883 rarr 119883 given by
119879119906 (119905) = int
119905
0
119906 (119904) 119889119904 (41)
Following [19] we see that the resolvent of 119879 is given by
119877 (120582 119879) 119906 (119905) =1
120582119906 (119905) +
1
1205822int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (42)
the Yosida approximation of 119879 is
119884 (120582 119879) 119906 (119905) = int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (43)
and the Mobius approximation of 119879 is
119860 (120582 119879) 119906 (119905) = (1 minus1
120582)int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (44)
Remark that for all 119906 isin C[0 119871] we have
|119879119906|119871 = max119905isin[0119871]
|119879119906 (119905)|
le max119905isin[0119871]
int
119905
0
|119906 (119904)| 119889119904
le max119905isin[0119871]
|119906 (119905)| int
119871
0
119889119904 = |119906|119871 sdot 119871
(45)
The above implies that 119879 is a contraction for 119871 le 1
The Scientific World Journal 5
If 119871 gt 1 then we can introduce for each 120576 gt 0 thefollowing norm onC[0 119871]
119906120576 = max119905isin[0119871]
119890119905120576|119906 (119905)| 119906 isin C [0 119871] (46)
Then a simple computation gives that
119879119906120576lt 120576119906120576 119906 isin C [0 119871] (47)
On the other hand
119906120576le |119906|119871 le 119890
119871120576119906120576 (48)
Remark that by Theorem 11 for all 120582 gt 1 we get
|119860 (120582 119879)|119871 = (120582 minus 1) (119890119879120582minus 1) le 1 (49)
if and only if |119879|119871le 1
It is clear that for estimating the powers of 119879 it seemsto be better to use the Yosida approximation or Mobiusapproximation than the resolvent approximation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the anonymous referees for theirvery careful reading and for useful suggestions that helpedin better exposing this material
References
[1] L D Lemle and L M Wu ldquoUniqueness of C0-semigroups
on a general locally convex vector space and an applicationrdquoSemigroup Forum vol 82 no 3 pp 485ndash496 2011
[2] R T Moore ldquoBanach algebras of operators on locally convexspacesrdquo Bulletin of the American Mathematical Society vol 75pp 68ndash73 1969
[3] F Pater ldquoProperties of multipliers on special algebras withapplication to signal processingrdquo Acta Technica NapocensisApplied Mathematics and Mechanics vol 54 pp 313ndash318 2011
[4] F Pater ldquoA multiplier algebra representation with applica-tion to harmonic signal modelsrdquo in Proceedings of the AIPInternational Conference on Numerical Analysis and AppliedMathematics vol 1479 pp 1075ndash1078 Kos Greece September2012
[5] M Kostic ldquoAbstract Volterra equations in locally convexspacesrdquo Science China Mathematics vol 55 no 9 pp 1797ndash1825 2012
[6] G R Allan ldquoA spectral theory for locally convex alebrasrdquoProceedings of the LondonMathematical Society vol 15 pp 399ndash421 1965
[7] L Waelbroeck Etude Spectrale des Algebres Completes vol 31Academie Royale de Belgique Classe des sciences MemoiresCol 1960
[8] L Waelbroeck ldquoAlgebres commutatives elements reguliersrdquoBulletin of the Belgian Mathematical Society vol 9 pp 42ndash491957
[9] E A Michael ldquoLocally multiplicatively-convex topologicalalgebrasrdquo Memoirs of the American Mathematical Society vol1952 no 11 79 pages 1952
[10] G A Joseph ldquoBoundedness and completeness in locally convexspaces and algebrasrdquo Journal of the Australian MathematicalSociety vol 24 no 1 pp 50ndash63 1977
[11] F G Bonales and R V Mendoza ldquoExtending the formula tocalculate the spectral radius of an operatorrdquo Proceedings of theAmerican Mathematical Society vol 126 no 1 pp 97ndash103 1998
[12] F Pater and T Binzar ldquoOn some ergodic theorems for a uni-versally bounded operatorrdquoCarpathian Journal of Mathematicsvol 26 no 1 pp 97ndash102 2010
[13] F Pater and L D Lemle ldquoOn some multiplication operatoralgebra problem with application to stochastic signal modelsrdquoin Proceedings of the International Conference on NumericalAnalysis and Applied Mathematics vol 1558 of AIP ConferenceProceedings pp 1661ndash1664 Rhodes Greece September 2013
[14] F Pater L D Lemle and T Binzar ldquoOn some Yosida typeapproximation theoremsrdquo in Proceedings of the InternationalConference on Numerical Analysis and Applied Mathematicsvol 1168 ofAIP Conference Proceedings pp 521ndash524 RethymnoCrete September 2009
[15] Y Katznelson and L Tzafriri ldquoOn power bounded operatorsrdquoJournal of Functional Analysis vol 68 no 3 pp 313ndash328 1986
[16] B Nagy and J Zemanek ldquoA resolvent condition implying powerboundednessrdquo Studia Mathematica vol 134 no 2 pp 143ndash1511999
[17] O Nevanlinna ldquoOn the growth of the resolvent operator forpower bounded operators Linear Operatorsrdquo Banach CenterPublications vol 38 pp 247ndash264 1997
[18] K Yosida ldquoOn the differentiability and the representation ofone-parameter semi-group of linear operatorsrdquo Journal of theMathematical Society of Japan vol 1 pp 15ndash21 1948
[19] O Nevanlinna Convergence of Iterations for Linear EquationsBirkhauser Basel Switzerland 1993
[20] A L Shields ldquoOn Mobius bounded operatorsrdquo Acta Universi-tatis Szegediensis Acta Scientiarum Mathematicarum vol 40no 3-4 pp 371ndash374 1978
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 3
Proof Let us suppose condition 119901[119877(120582 119879)] le 119862(|120582| minus 1) istrue for all 119901 isin P for any 119896 isin Nlowast and |120582| gt 1 For 119896 isin Nlowast
fixed by choosing the integration path Γ |120582| = 1 + 1119896 withthe aid of the functional calculus from the algebra (119876P(119883))0we obtain
119879119896=1
2120587119894intΓ
120582119896119877 (120582 119879) 119889120582 (lowast)
Thus for all 119901 isin P we have
119901 (119879119896) le
1
2120587intΓ
|120582|119896119901 (119877 (120582 119879)) 119889120582
le1
2120587sdotmax120582isinΓ
|120582|119896sdotmax120582isinΓ
119862
|120582| minus 1sdot intΓ
119889120582
le1
2120587sdot (1 +
1
119896)
119896
sdot 119862119896 sdot 2120587 (1 +1
119896) le 119862119890 (119896 + 1)
(17)
which implies the desired resultMoreover we can formulate the following
Theorem 5 If 119879 isin (119876P(119883))0 and
sup119901isinP
119901 [119877(120582 119879)119896] le
119862
(|120582| minus 1)119896 (18)
for 119896 isin Nlowast and for all 120582 isin C with |120582| gt 1 then
sup119901isinP
119901 (119879119896) le 119862
119896119890119896
119896119896le 119862radic2120587 (119896 + 1) 119896 isin N
lowast (19)
Proof Integrating (lowast) by parts 119895minus1 times for 119895 gt 2 we obtain
119879119896=(minus1)119895minus1
2120587119894intΓ
(119895 + 1)120582119896+119895minus1
(119896 + 1) sdot sdot sdot (119896 + 119895 minus 1)119877(120582 119879)
119895119889120582 (20)
Now choosing Γ the circle of radius 1 + 119895119896 and by usingthe hypothesis for 119895 rarr infin we get
sup119901isinP
119901 (119879119896) le 119862
119896119890119896
119896119896le 119862radic2120587 (119896 + 1) (21)
The last inequality was obtained by using Stirlingrsquos approxi-mation
Now for 119879 isin (119876P(119883))0 we introduce (see [18]) thefollowing
Definition 6 The Yosida approximation 119884(120582 119879) of 119879 for 120582 isin120588119882(119879) cap C is defined as
119884 (120582 119879) = 120582119879119877 (120582 119879) (22)
Next theorem shows how an operator from the (119876P(119883))0
algebra is related to its Yosida approximation
Theorem 7 The Yosida approximation 119884(120582 119879) is analytic for120582 isin 120588119882(119879) cap C and the series representation
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895(23)
converges for |120582| gt 119903P(119879) Moreover
(1) 119884(120582 119879) = 1205822119877(120582 119879) minus 120582119868(2) if there exists 119901 isin P such that 119903P(119879) lt 119901(119879) then
119901 (119884 (120582 119879) minus 119879) le119901 (1198792)
|120582| minus 119901 (119879) (24)
for |120582| gt 119901(119879)(3) 120590119882(119884(120582 119879)) = 119911(1 minus 119911120582) 119911 isin 120590
119882(119879)
Proof By evaluating119884(120582 119879) in terms of the resolvent119877(120582 119879)for |120582| gt 119903P(119879) we obtain
119884 (120582 119879) = 120582119879119877 (120582 119879) = 120582119879(120582119868 minus 119879)minus1
= 120582119879 sdot
infin
sum
119895=0
119879119895
120582119895+1=
infin
sum
119895=0
119879119895+1
120582119895
(25)
fromwhere it follows that the assertion of the theorem is trueMoreover
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895= 120582119868 + 119879 +
1198792
120582+ sdot sdot sdot +
119879119899+1
120582119899+ sdot sdot sdot minus 120582119868
= 1205822
infin
sum
119895=0
119879119895
120582119895+1minus 120582119868 = 120582
2119877 (120582 119879) minus 120582119868
(26)
so (1) is trueTo prove (2) one can observe that from
119884 (120582 119879) =
infin
sum
119895=0
119879119895+1
120582119895 (27)
it follows that
119884 (120582 119879) minus 119879 =
infin
sum
119895=0
1198792
120582(119879119895
120582119895) (28)
on a set for which |120582| gt 119903P(119879) Moreover
119901 (119884 (120582 119879) minus 119879) le
infin
sum
119895=0
119901(1198792
120582)119901(
119879119895
120582119895)
le 119901(1198792
120582) sdot
infin
sum
119895=0
119901(119879
120582)
119895
= 119901(1198792
120582)
1
1 minus 119901 (119879120582)=
119901 (1198792)
|120582| minus 119901 (119879)
(29)
for |120582| gt 119901(119879) gt 119903P(119879)A simple reasoning shows that 119877(120582 119879) isin (119876P(119883))0 then
it follows 119884(120582 119879) isin (119876P(119883))0From [19 Theorem 3114] for |120582| gt |119911| we have
120590119882(119884 (120582 119879)) = 119884 (120582 120590
119882(119879)) (30)
for all 119911 isin 120590119882(119879) and
119884 (120582 119911) =
infin
sum
119895=0
119911119895+1
120582119895(31)
on |120582| gt |119911| which could be written as 119884(120582 119911) = 119911(1 minus 119911120582)for any 119911 isin 120590
119882(119879) so (3) is proved
4 The Scientific World Journal
Below we state an equivalence between a power boundedoperator from the (119876P(119883))0 algebra and the power of itsYosida approximation
Theorem 8 Let119879 isin (119876P(119883))0 and119884(120582 119879) its Yosida approx-imation Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119888 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119884(120582 119879)
119896) le 119888(1 minus 1|120582|)
119896 for any 119896 isin Nlowast
and for all 120582 isin C with |120582| gt 1
Proof Property (i) implies 119903P(119879) le 1 so that the argumenta-tion given in the proof of Theorem 7 implies that any 120582 isin C
with |120582| gt 1 belongs to the resolvent set of 119879 Hence usingthe generalized binomial formula we get
119884(120582 119879)119896=
infin
sum
119895=0
(119896 + 119895 minus 1
119895)119879119895+119896
120582119895 (32)
Now by applying (i) again we obtain
119901 (119884(120582 119879)119896) le 119888
infin
sum
119895=0
(119896 + 119895 minus 1
119895)(
1
|120582|)
119895
=119888
(1 minus 1 |120582|)119896
(33)
for any 119901 isin P whence by passing to supremum theinequality (ii) holds
Conversely (i) is a direct consequence of (ii)
For 120583 isin 120588119882(119879) consider now the following Mobius
transformation (see [20])
120595120582(120583) =
(120582 minus 1) 120583
120582 minus 120583 if 120582 = infin
120583 if 120582 = infin(34)
Definition 9 TheMobius approximation of 119879 is defined as
119860 (120582 119879) = 120595120582(119879) (35)
Proposition 10 119860(120582 119879) is holomorphic in 120582 isin 120588119882(119879) cap C
and satisfies
119860 (120582 119879) = (1 minus1
120582)119884 (120582 119879) 120582 = 0 (36)
Proof Let 120582 isin 120588119882(119879) cap C 0 By evaluating the right
member of the above equality we get successively
(1 minus1
120582)119884 (120582 119879)
= (1 minus1
120582) 120582119879119877 (120582 119879) = (120582 minus 1) 119879119877 (120582 119879)
=(120582 minus 1) 119879
120582119868 minus 119879= 119860 (120582 119879)
(37)
for 120582 = infin If 120582 = infin then from Definition 9 we have119860(120582 119879) = 119879 On the other side (1 minus 1120582)119884(120582 119879) convergesto 119879 when 120582 rarr infin
A similar result as in Theorem 8 is given below
Theorem 11 Let 119879 isin (119876P(119883))0 and 119860(120582 119879) its approxima-tion as above Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119862 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119860(120582 119879)
119896) le 119862 for any 119896 isin Nlowast and for every
120582 isin C with |120582| gt 1
Proof FromTheorem 8 for 119879 isin (119876P(119883))0
sup119901isinP
119901 (119879119896) le 119862 (38)
is equivalent to
sup119901isinP
119901 (119884(120582 119879)119896) le
119862
(1 minus 1 |120582|)119896 (39)
The conclusion follows taking into account that
119860(120582 119879)119896= (1 minus
1
120582)
119896
sdot 119884(120582 119879)119896 (40)
for 119896 isin Nlowast
4 Application
For 119871 gt 0 let 119883 = C[0 119871] be the space of continuous func-tions on [0 119871] endowed with the norm |119906|
119871= max
[0119871]|119906(119905)|
Consider 119879 119883 rarr 119883 given by
119879119906 (119905) = int
119905
0
119906 (119904) 119889119904 (41)
Following [19] we see that the resolvent of 119879 is given by
119877 (120582 119879) 119906 (119905) =1
120582119906 (119905) +
1
1205822int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (42)
the Yosida approximation of 119879 is
119884 (120582 119879) 119906 (119905) = int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (43)
and the Mobius approximation of 119879 is
119860 (120582 119879) 119906 (119905) = (1 minus1
120582)int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (44)
Remark that for all 119906 isin C[0 119871] we have
|119879119906|119871 = max119905isin[0119871]
|119879119906 (119905)|
le max119905isin[0119871]
int
119905
0
|119906 (119904)| 119889119904
le max119905isin[0119871]
|119906 (119905)| int
119871
0
119889119904 = |119906|119871 sdot 119871
(45)
The above implies that 119879 is a contraction for 119871 le 1
The Scientific World Journal 5
If 119871 gt 1 then we can introduce for each 120576 gt 0 thefollowing norm onC[0 119871]
119906120576 = max119905isin[0119871]
119890119905120576|119906 (119905)| 119906 isin C [0 119871] (46)
Then a simple computation gives that
119879119906120576lt 120576119906120576 119906 isin C [0 119871] (47)
On the other hand
119906120576le |119906|119871 le 119890
119871120576119906120576 (48)
Remark that by Theorem 11 for all 120582 gt 1 we get
|119860 (120582 119879)|119871 = (120582 minus 1) (119890119879120582minus 1) le 1 (49)
if and only if |119879|119871le 1
It is clear that for estimating the powers of 119879 it seemsto be better to use the Yosida approximation or Mobiusapproximation than the resolvent approximation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the anonymous referees for theirvery careful reading and for useful suggestions that helpedin better exposing this material
References
[1] L D Lemle and L M Wu ldquoUniqueness of C0-semigroups
on a general locally convex vector space and an applicationrdquoSemigroup Forum vol 82 no 3 pp 485ndash496 2011
[2] R T Moore ldquoBanach algebras of operators on locally convexspacesrdquo Bulletin of the American Mathematical Society vol 75pp 68ndash73 1969
[3] F Pater ldquoProperties of multipliers on special algebras withapplication to signal processingrdquo Acta Technica NapocensisApplied Mathematics and Mechanics vol 54 pp 313ndash318 2011
[4] F Pater ldquoA multiplier algebra representation with applica-tion to harmonic signal modelsrdquo in Proceedings of the AIPInternational Conference on Numerical Analysis and AppliedMathematics vol 1479 pp 1075ndash1078 Kos Greece September2012
[5] M Kostic ldquoAbstract Volterra equations in locally convexspacesrdquo Science China Mathematics vol 55 no 9 pp 1797ndash1825 2012
[6] G R Allan ldquoA spectral theory for locally convex alebrasrdquoProceedings of the LondonMathematical Society vol 15 pp 399ndash421 1965
[7] L Waelbroeck Etude Spectrale des Algebres Completes vol 31Academie Royale de Belgique Classe des sciences MemoiresCol 1960
[8] L Waelbroeck ldquoAlgebres commutatives elements reguliersrdquoBulletin of the Belgian Mathematical Society vol 9 pp 42ndash491957
[9] E A Michael ldquoLocally multiplicatively-convex topologicalalgebrasrdquo Memoirs of the American Mathematical Society vol1952 no 11 79 pages 1952
[10] G A Joseph ldquoBoundedness and completeness in locally convexspaces and algebrasrdquo Journal of the Australian MathematicalSociety vol 24 no 1 pp 50ndash63 1977
[11] F G Bonales and R V Mendoza ldquoExtending the formula tocalculate the spectral radius of an operatorrdquo Proceedings of theAmerican Mathematical Society vol 126 no 1 pp 97ndash103 1998
[12] F Pater and T Binzar ldquoOn some ergodic theorems for a uni-versally bounded operatorrdquoCarpathian Journal of Mathematicsvol 26 no 1 pp 97ndash102 2010
[13] F Pater and L D Lemle ldquoOn some multiplication operatoralgebra problem with application to stochastic signal modelsrdquoin Proceedings of the International Conference on NumericalAnalysis and Applied Mathematics vol 1558 of AIP ConferenceProceedings pp 1661ndash1664 Rhodes Greece September 2013
[14] F Pater L D Lemle and T Binzar ldquoOn some Yosida typeapproximation theoremsrdquo in Proceedings of the InternationalConference on Numerical Analysis and Applied Mathematicsvol 1168 ofAIP Conference Proceedings pp 521ndash524 RethymnoCrete September 2009
[15] Y Katznelson and L Tzafriri ldquoOn power bounded operatorsrdquoJournal of Functional Analysis vol 68 no 3 pp 313ndash328 1986
[16] B Nagy and J Zemanek ldquoA resolvent condition implying powerboundednessrdquo Studia Mathematica vol 134 no 2 pp 143ndash1511999
[17] O Nevanlinna ldquoOn the growth of the resolvent operator forpower bounded operators Linear Operatorsrdquo Banach CenterPublications vol 38 pp 247ndash264 1997
[18] K Yosida ldquoOn the differentiability and the representation ofone-parameter semi-group of linear operatorsrdquo Journal of theMathematical Society of Japan vol 1 pp 15ndash21 1948
[19] O Nevanlinna Convergence of Iterations for Linear EquationsBirkhauser Basel Switzerland 1993
[20] A L Shields ldquoOn Mobius bounded operatorsrdquo Acta Universi-tatis Szegediensis Acta Scientiarum Mathematicarum vol 40no 3-4 pp 371ndash374 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 The Scientific World Journal
Below we state an equivalence between a power boundedoperator from the (119876P(119883))0 algebra and the power of itsYosida approximation
Theorem 8 Let119879 isin (119876P(119883))0 and119884(120582 119879) its Yosida approx-imation Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119888 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119884(120582 119879)
119896) le 119888(1 minus 1|120582|)
119896 for any 119896 isin Nlowast
and for all 120582 isin C with |120582| gt 1
Proof Property (i) implies 119903P(119879) le 1 so that the argumenta-tion given in the proof of Theorem 7 implies that any 120582 isin C
with |120582| gt 1 belongs to the resolvent set of 119879 Hence usingthe generalized binomial formula we get
119884(120582 119879)119896=
infin
sum
119895=0
(119896 + 119895 minus 1
119895)119879119895+119896
120582119895 (32)
Now by applying (i) again we obtain
119901 (119884(120582 119879)119896) le 119888
infin
sum
119895=0
(119896 + 119895 minus 1
119895)(
1
|120582|)
119895
=119888
(1 minus 1 |120582|)119896
(33)
for any 119901 isin P whence by passing to supremum theinequality (ii) holds
Conversely (i) is a direct consequence of (ii)
For 120583 isin 120588119882(119879) consider now the following Mobius
transformation (see [20])
120595120582(120583) =
(120582 minus 1) 120583
120582 minus 120583 if 120582 = infin
120583 if 120582 = infin(34)
Definition 9 TheMobius approximation of 119879 is defined as
119860 (120582 119879) = 120595120582(119879) (35)
Proposition 10 119860(120582 119879) is holomorphic in 120582 isin 120588119882(119879) cap C
and satisfies
119860 (120582 119879) = (1 minus1
120582)119884 (120582 119879) 120582 = 0 (36)
Proof Let 120582 isin 120588119882(119879) cap C 0 By evaluating the right
member of the above equality we get successively
(1 minus1
120582)119884 (120582 119879)
= (1 minus1
120582) 120582119879119877 (120582 119879) = (120582 minus 1) 119879119877 (120582 119879)
=(120582 minus 1) 119879
120582119868 minus 119879= 119860 (120582 119879)
(37)
for 120582 = infin If 120582 = infin then from Definition 9 we have119860(120582 119879) = 119879 On the other side (1 minus 1120582)119884(120582 119879) convergesto 119879 when 120582 rarr infin
A similar result as in Theorem 8 is given below
Theorem 11 Let 119879 isin (119876P(119883))0 and 119860(120582 119879) its approxima-tion as above Then the following assertions are equivalent
(i) sup119901isinP119901(119879
119896) le 119862 for any 119896 isin Nlowast
(ii) sup119901isinP119901(119860(120582 119879)
119896) le 119862 for any 119896 isin Nlowast and for every
120582 isin C with |120582| gt 1
Proof FromTheorem 8 for 119879 isin (119876P(119883))0
sup119901isinP
119901 (119879119896) le 119862 (38)
is equivalent to
sup119901isinP
119901 (119884(120582 119879)119896) le
119862
(1 minus 1 |120582|)119896 (39)
The conclusion follows taking into account that
119860(120582 119879)119896= (1 minus
1
120582)
119896
sdot 119884(120582 119879)119896 (40)
for 119896 isin Nlowast
4 Application
For 119871 gt 0 let 119883 = C[0 119871] be the space of continuous func-tions on [0 119871] endowed with the norm |119906|
119871= max
[0119871]|119906(119905)|
Consider 119879 119883 rarr 119883 given by
119879119906 (119905) = int
119905
0
119906 (119904) 119889119904 (41)
Following [19] we see that the resolvent of 119879 is given by
119877 (120582 119879) 119906 (119905) =1
120582119906 (119905) +
1
1205822int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (42)
the Yosida approximation of 119879 is
119884 (120582 119879) 119906 (119905) = int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (43)
and the Mobius approximation of 119879 is
119860 (120582 119879) 119906 (119905) = (1 minus1
120582)int
119905
0
119890(119905minus119904)120582
119906 (119904) 119889119904 (44)
Remark that for all 119906 isin C[0 119871] we have
|119879119906|119871 = max119905isin[0119871]
|119879119906 (119905)|
le max119905isin[0119871]
int
119905
0
|119906 (119904)| 119889119904
le max119905isin[0119871]
|119906 (119905)| int
119871
0
119889119904 = |119906|119871 sdot 119871
(45)
The above implies that 119879 is a contraction for 119871 le 1
The Scientific World Journal 5
If 119871 gt 1 then we can introduce for each 120576 gt 0 thefollowing norm onC[0 119871]
119906120576 = max119905isin[0119871]
119890119905120576|119906 (119905)| 119906 isin C [0 119871] (46)
Then a simple computation gives that
119879119906120576lt 120576119906120576 119906 isin C [0 119871] (47)
On the other hand
119906120576le |119906|119871 le 119890
119871120576119906120576 (48)
Remark that by Theorem 11 for all 120582 gt 1 we get
|119860 (120582 119879)|119871 = (120582 minus 1) (119890119879120582minus 1) le 1 (49)
if and only if |119879|119871le 1
It is clear that for estimating the powers of 119879 it seemsto be better to use the Yosida approximation or Mobiusapproximation than the resolvent approximation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the anonymous referees for theirvery careful reading and for useful suggestions that helpedin better exposing this material
References
[1] L D Lemle and L M Wu ldquoUniqueness of C0-semigroups
on a general locally convex vector space and an applicationrdquoSemigroup Forum vol 82 no 3 pp 485ndash496 2011
[2] R T Moore ldquoBanach algebras of operators on locally convexspacesrdquo Bulletin of the American Mathematical Society vol 75pp 68ndash73 1969
[3] F Pater ldquoProperties of multipliers on special algebras withapplication to signal processingrdquo Acta Technica NapocensisApplied Mathematics and Mechanics vol 54 pp 313ndash318 2011
[4] F Pater ldquoA multiplier algebra representation with applica-tion to harmonic signal modelsrdquo in Proceedings of the AIPInternational Conference on Numerical Analysis and AppliedMathematics vol 1479 pp 1075ndash1078 Kos Greece September2012
[5] M Kostic ldquoAbstract Volterra equations in locally convexspacesrdquo Science China Mathematics vol 55 no 9 pp 1797ndash1825 2012
[6] G R Allan ldquoA spectral theory for locally convex alebrasrdquoProceedings of the LondonMathematical Society vol 15 pp 399ndash421 1965
[7] L Waelbroeck Etude Spectrale des Algebres Completes vol 31Academie Royale de Belgique Classe des sciences MemoiresCol 1960
[8] L Waelbroeck ldquoAlgebres commutatives elements reguliersrdquoBulletin of the Belgian Mathematical Society vol 9 pp 42ndash491957
[9] E A Michael ldquoLocally multiplicatively-convex topologicalalgebrasrdquo Memoirs of the American Mathematical Society vol1952 no 11 79 pages 1952
[10] G A Joseph ldquoBoundedness and completeness in locally convexspaces and algebrasrdquo Journal of the Australian MathematicalSociety vol 24 no 1 pp 50ndash63 1977
[11] F G Bonales and R V Mendoza ldquoExtending the formula tocalculate the spectral radius of an operatorrdquo Proceedings of theAmerican Mathematical Society vol 126 no 1 pp 97ndash103 1998
[12] F Pater and T Binzar ldquoOn some ergodic theorems for a uni-versally bounded operatorrdquoCarpathian Journal of Mathematicsvol 26 no 1 pp 97ndash102 2010
[13] F Pater and L D Lemle ldquoOn some multiplication operatoralgebra problem with application to stochastic signal modelsrdquoin Proceedings of the International Conference on NumericalAnalysis and Applied Mathematics vol 1558 of AIP ConferenceProceedings pp 1661ndash1664 Rhodes Greece September 2013
[14] F Pater L D Lemle and T Binzar ldquoOn some Yosida typeapproximation theoremsrdquo in Proceedings of the InternationalConference on Numerical Analysis and Applied Mathematicsvol 1168 ofAIP Conference Proceedings pp 521ndash524 RethymnoCrete September 2009
[15] Y Katznelson and L Tzafriri ldquoOn power bounded operatorsrdquoJournal of Functional Analysis vol 68 no 3 pp 313ndash328 1986
[16] B Nagy and J Zemanek ldquoA resolvent condition implying powerboundednessrdquo Studia Mathematica vol 134 no 2 pp 143ndash1511999
[17] O Nevanlinna ldquoOn the growth of the resolvent operator forpower bounded operators Linear Operatorsrdquo Banach CenterPublications vol 38 pp 247ndash264 1997
[18] K Yosida ldquoOn the differentiability and the representation ofone-parameter semi-group of linear operatorsrdquo Journal of theMathematical Society of Japan vol 1 pp 15ndash21 1948
[19] O Nevanlinna Convergence of Iterations for Linear EquationsBirkhauser Basel Switzerland 1993
[20] A L Shields ldquoOn Mobius bounded operatorsrdquo Acta Universi-tatis Szegediensis Acta Scientiarum Mathematicarum vol 40no 3-4 pp 371ndash374 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 5
If 119871 gt 1 then we can introduce for each 120576 gt 0 thefollowing norm onC[0 119871]
119906120576 = max119905isin[0119871]
119890119905120576|119906 (119905)| 119906 isin C [0 119871] (46)
Then a simple computation gives that
119879119906120576lt 120576119906120576 119906 isin C [0 119871] (47)
On the other hand
119906120576le |119906|119871 le 119890
119871120576119906120576 (48)
Remark that by Theorem 11 for all 120582 gt 1 we get
|119860 (120582 119879)|119871 = (120582 minus 1) (119890119879120582minus 1) le 1 (49)
if and only if |119879|119871le 1
It is clear that for estimating the powers of 119879 it seemsto be better to use the Yosida approximation or Mobiusapproximation than the resolvent approximation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the anonymous referees for theirvery careful reading and for useful suggestions that helpedin better exposing this material
References
[1] L D Lemle and L M Wu ldquoUniqueness of C0-semigroups
on a general locally convex vector space and an applicationrdquoSemigroup Forum vol 82 no 3 pp 485ndash496 2011
[2] R T Moore ldquoBanach algebras of operators on locally convexspacesrdquo Bulletin of the American Mathematical Society vol 75pp 68ndash73 1969
[3] F Pater ldquoProperties of multipliers on special algebras withapplication to signal processingrdquo Acta Technica NapocensisApplied Mathematics and Mechanics vol 54 pp 313ndash318 2011
[4] F Pater ldquoA multiplier algebra representation with applica-tion to harmonic signal modelsrdquo in Proceedings of the AIPInternational Conference on Numerical Analysis and AppliedMathematics vol 1479 pp 1075ndash1078 Kos Greece September2012
[5] M Kostic ldquoAbstract Volterra equations in locally convexspacesrdquo Science China Mathematics vol 55 no 9 pp 1797ndash1825 2012
[6] G R Allan ldquoA spectral theory for locally convex alebrasrdquoProceedings of the LondonMathematical Society vol 15 pp 399ndash421 1965
[7] L Waelbroeck Etude Spectrale des Algebres Completes vol 31Academie Royale de Belgique Classe des sciences MemoiresCol 1960
[8] L Waelbroeck ldquoAlgebres commutatives elements reguliersrdquoBulletin of the Belgian Mathematical Society vol 9 pp 42ndash491957
[9] E A Michael ldquoLocally multiplicatively-convex topologicalalgebrasrdquo Memoirs of the American Mathematical Society vol1952 no 11 79 pages 1952
[10] G A Joseph ldquoBoundedness and completeness in locally convexspaces and algebrasrdquo Journal of the Australian MathematicalSociety vol 24 no 1 pp 50ndash63 1977
[11] F G Bonales and R V Mendoza ldquoExtending the formula tocalculate the spectral radius of an operatorrdquo Proceedings of theAmerican Mathematical Society vol 126 no 1 pp 97ndash103 1998
[12] F Pater and T Binzar ldquoOn some ergodic theorems for a uni-versally bounded operatorrdquoCarpathian Journal of Mathematicsvol 26 no 1 pp 97ndash102 2010
[13] F Pater and L D Lemle ldquoOn some multiplication operatoralgebra problem with application to stochastic signal modelsrdquoin Proceedings of the International Conference on NumericalAnalysis and Applied Mathematics vol 1558 of AIP ConferenceProceedings pp 1661ndash1664 Rhodes Greece September 2013
[14] F Pater L D Lemle and T Binzar ldquoOn some Yosida typeapproximation theoremsrdquo in Proceedings of the InternationalConference on Numerical Analysis and Applied Mathematicsvol 1168 ofAIP Conference Proceedings pp 521ndash524 RethymnoCrete September 2009
[15] Y Katznelson and L Tzafriri ldquoOn power bounded operatorsrdquoJournal of Functional Analysis vol 68 no 3 pp 313ndash328 1986
[16] B Nagy and J Zemanek ldquoA resolvent condition implying powerboundednessrdquo Studia Mathematica vol 134 no 2 pp 143ndash1511999
[17] O Nevanlinna ldquoOn the growth of the resolvent operator forpower bounded operators Linear Operatorsrdquo Banach CenterPublications vol 38 pp 247ndash264 1997
[18] K Yosida ldquoOn the differentiability and the representation ofone-parameter semi-group of linear operatorsrdquo Journal of theMathematical Society of Japan vol 1 pp 15ndash21 1948
[19] O Nevanlinna Convergence of Iterations for Linear EquationsBirkhauser Basel Switzerland 1993
[20] A L Shields ldquoOn Mobius bounded operatorsrdquo Acta Universi-tatis Szegediensis Acta Scientiarum Mathematicarum vol 40no 3-4 pp 371ndash374 1978
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of