multipliers, linear functionals and the frЙchet

transactions of theamerican mathematical societyVolume 322, Number 2, December 1990

MULTIPLIERS, LINEAR FUNCTIONALS AND

THE FRÉCHET ENVELOPE OF THE SMIRNOV CLASS NfV")

MAREK NAWROCKI

Abstract. Linear topological properties of the Smirnov class A/,(U") of the

unitpolydisk U" in C" are studied. All multipliers of A/.ÍU") into the Hardy

spaces H_(U"), 0 < p < oo , are described. A representation of the continuous

linear functionals on Nt(Vn) is obtained. The Fréchet envelope of A.fU") is

constructed. It is proved that if n > 1 , then Nt(Vn) is not isomorphic to

tf.(U').

1. Introduction

The Smirnov classes are well-known spaces of analytic functions that arise

naturally in many contexts of the geometric theory of H -spaces. The Smirnov

class Nt of the unit disk in the complex plane was extensively studied by Yana-

gihara [16, 17], who described all continuous linear functionals on Nt and

found all multipliers of Nt into Hardy spaces. It can be observed that the

crucial step in the proofs of Yanagihara's results are the best possible estimates

of the Taylor coefficients of functions in Nm. See [18], Stoll [15] obtained

analogues of Yanagihara's estimates for functions in the Smirnov class N:t(D)

of an arbitrary irreducible bounded symmetric domain D in Cn, which is best

possible if D is the unit ball in C" (see [8]). The present paper is a study of

the topological vector space structure of the Smirnov class NfV") of the unit

polydisk U" in C" .

The paper is organized as follows. §2 contains preliminary definitions and

notation. In §3, we study the mean growth of the Taylor coefficients of functions

from the class Nt(U"). This leads us to the construction of a nuclear Fréchet

sequence space M[ß], which is basic for the rest of the paper.

In §4 we prove our main result (Theorem 4.5), which describes all multipliers

of NfU") into the Hardy spaces of U" . This result is applied in §5 to obtain a

representation of the continuous linear functionals on Nt(Vn). It turns out that

the dual space of NfVn) can be identified with the dual of M[ß]. This implies

Received by the editors March 21, 1988 and, in revised form, December 5, 1988.

1980 Mathematics Subject Classification (1985 Revision). Primary 32A22, 32A35, 46A06; Sec-ondary 32A05, 46A12.

Key words and phrases. Smirnov class, multipliers, Fréchet envelopes, linear functionals, nuclear

spaces.

© 1990 American Mathematical Society

0002-9947/90 $1.00+$.25 per page

493

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

494 MAREK NAWROCKI

that M[ß] is the so-called Fréchet envelope of A^U"), i.e., the completion of

Nt(Vn) equipped with the strongest locally convex topology on Nt(Vn) which

is weaker than the original topology of Nt(U"). We show that the Fréchet

envelope of Nt(V ) is not isomorphic to the Fréchet envelope of any class

Nt(V"), n > 1. In consequence, Nt(U ) is not isomorphic to At(U") if

n > 1 .

2. Preliminaries

Throughout this paper, U will denote the open ur't disk in the complex

plane C, T the unit circle, I = (0, 1) the unit interval, dm the normalized

Lebesge measure on T, and Z+ the set of all nonnegative integers. Moreover,

for a natural number n, U", T", l", dm , and if will denote the «-foldn ' +

products of U, T, I, dm , and Z+ , respectively.

We will denote by H(V") the space of all analytic functions on U" endowed

with the compact-open topology k .

If f G H(V") and a = (a,, ... , an) G Z" , then the ath Taylor coefficient

of / will be denoted by fi(a).

If z = (zx,...,zfeC", r = (rx,...,rn)ef, a = (a,, ... , a„) E Zn+,

s e I, the following abbreviations will be used: rz := (rxzx, ... , rnzf , sz :=

(szx, ... ,szf, z := zf---zf.A function fi G H(V") is in the Nevanlinna class A/(U") provided that

sup / log+ \f(rco)\ dmfco) < oo.0<r<l JT

The Smirnov class Nt(V") is the subspace of N(U") consisting of those /

for which the family {log+ \fr\: r G 1} is uniformly integrable on T" , where

ffoj) = f(roj).It is well known that the functional

||/||= sup [ log(l + |/r|)rfmn0<r<l JT

is a complete F-norm on AJU"), and so the balls B(e) = {fi G NfVn): \\f\\ <

e}, e > 0, are a base of neighborhoods of zero for a complete metrizable

vector topology, v, on NJAff). Thus, (NfV"),u) is an F-space. Moreover,

for each / G NfV") the radial limits limr^,- fi(rco) = f*(oj) exist for almost

all co gT" and \\fi\\ = jr log(l + \f\)dmn . The reader is referred to [11] for

information on Nm(Vn).

The same arguments as in the case n = 1 (we use the «-subharmonity of

log(l + l/D) show that

(2-11 «l+wsîiiFki

for all / G NfV") and z = (z,, ... , z ) G U" . This estimate suggests a study


LINEAR FUNCTIONALS AND THE FRECHET ENVELOPE 495

of the space FfV") of all analytic functions / on U" for which

||/lf = sup |/(z)|exp l-cflil - |z,|r' ) < oo*eu" V ti /

for all c > 0. It is easy to see that F^(Vn) endowed with the topology deter-

mined by the family of norms {|| • ||c : c > 0} is a Fréchet space (locally convex

F-space). Note that F+ = FfVv) is the Fréchet envelope of N+ = AJU1)

invented by Yanagihara [16] (see also [15, §6]).

As a simple consequence of inequality (2.1), we obtain

Proposition 2.1. Nt(Vn) is contained in FfVn), and the inclusion mapping is

continuous.

3. Taylor coefficients

If ß = [ßm(a): m G N, a G iff] is a matrix whose entries are positive

numbers, then a family x = {x(a)}al£Z* of complex numbers is in the Köthe

space M[ß] provided that

(3.0) ||jr|| = sup \x(a)\ß ia) < ooa€Z"

for all m G N . M[ß] equipped with the topology determined by the sequence

of norms {|| • ||m: m G N} is a Fréchet space (see [10, p. 17]).

Proposition 3.1 [10, Proposition 7.4.8]. If the matrix ß — [ßm{a)] satisfies

i3-1) J2ßmia)/ßm+iia)<°° for each m Gn,a

then:

(a) Each continuous linear functional T defined on the space M[ß] is of the

formTx = ^X(a)x(a), x = {x(a)} G M[ß],

a

where k = {1(a)} a&„ is a family of complex numbers such that

sup \k(a)\l ßm(a) < co for some m GN.

(b) The space M[ß] is nuclear.

In the sequel, we define and then we fix for the rest of the paper a very special

matrix ß = [ßm(a)]. In §5, we prove that the Köthe space M[ß] defined by ß

is isomorphic to the Fréchet envelope of NfVn).

If a G Z" we let a* = (a*, ... , off) be the nonincreasing rearrangement of

a and definei \ i * * i~,rn, 1/(7+1)

amjia):=ia\ ••■••aJ/2 )

where m GN , 1 < j < n . Moreover, we define

(3.2) LJa) :=max{k g {1 , ... , n}: am Ja)/a < 1/2 for all y < k} ,


496 MAREK NAWROCKI

where a G Z" \{0}, m G N, 0/0=1. The integer Lm(a) will be called the

m th essential length of a. If we take

(3-3) amia)-am,LJa)ia)>

then it is clear that

(3.4) am(a)/a]< 1/2 for all j < LJa),

(3.5) aJ<2V2am(a) for all j > Lm(a).

It is obvious that Lx(a) < Lfa) < ■■■ <n for each a G Z"\{0} .

Finally, we define

,.,, , , . (2-mam(a) ifaGl"+\{0},

(3-6) *«(") = \ ~-m -, n[2 if a = 0

and

(3.7) ßm{a):=exp{-bja))

for all q e Z" and m G N.

Lemma 3.2. (a)

« rww^J2'72 ^»<l-h-i(«).

[2 //Lm(Q) = Lm+,(Q)

/ora// meN, a G Z" \{0} ;

(b) a„+m+/(a) < 2n/2'l¡{n+l)aja) for all l, m eN, « e Z"+\{0} ;

(c) èm+1(a)/ôm(a)<2-1/2 /ora// m G N, aeZ"+\{0};

(d) Enßja)/ßm+f«)<™ fior each m GN.

Proofi (a) If Lm(«) = Lm+X{a) =: k, then am+l(a)/am(a) = 2~1/('c+1) <

2-i/(»+i)

Suppose now Lm{a) < Lm+]{a) =: i. Then, using (3.4) and (3.5), we obtain

am+x{a)/am(a)<a*/2am(a)<2l/2.

(c) is an immediate consequence of (a), while (b) follows from (a) and the

fact that the set {m G N: Lm{a) < Lm+X{a)} has no more than n elements.

(d) we have

ßm{a)/ßm+x{a)=exp{-bm{a) + bm+x{a))

<exp(-(l-2-*/2)¿>,»).

Therefore, for the proof of (d), it is enough to show that

n

Y Y exp(-c(,v--a¡)'/(A'+1))<oo for all c> 0.k=l L<<t)=k


LINEAR FUNCTIONALS AND THE FRÉCHET ENVELOPE 497

However,

Ei i * *A/(k+l).exp(-c(a, ■■•atk)' )

Lja)=k

<k\ Y exp(-c(ai---a,)1M+1))

at>-->ak>l

oo^ , , \~* k , l/(fc+l)s<k\ 2_^ax exp(-ca, ) < oo.

The proof is complete.

Proposition 3.3. The mapping fi -^ {/(a)} is a continuous embedding of Ft (U")

into M[ß].

In §5 we show that in fact t is a topological linear isomorphism of FfUn)

onto M[ß].

Proof. For a moment, fix q G Z" \{0}, m G N, and an arbitrary function

fiGH(Un) satisfying

(3.8) !/(z)|<exp(2-wn(1-l27l)"1 > zeU"'

For the sake of convenience, let us assume that a, > a2 > • ■ • > an . It is well

known that for all r = (rx, ... , rf Gl"

\fi(«)\ < r;"'---r;"" max{|/(z,, ... , zf\: |z,| = rJt j = 1,... , «}.

Therefore, if r G f then

|/(a)|<r-a'--T;a»expÍ2-Mn(l-0)

Take r} := 1 - x}, where Xj := am{a)/aj for ;' = 1,2, ... , Lm{a) =: k

and X: = 1/2 for j > Lm{a) (see (3.4)). Using the inequality 1 - x > c~2x ,

0 < x < 1/2, we obtain:

( " 2"'" A|/(a)|<exp \2j2ajXj +

j=\ In

However, by (3.4) and (3.5),

n -, — m n ¡ i~.

2 L ajxj + v~av = 2kaJa) + L aj +2j=\ x\'"xn J=k+X am(a)

k

< (2k + (n- k)2V2 + 2"-k)aJa) < Aaja),

where A is an absolute constant. Therefore, we have proved that

(3.9) \(fa)\<exp{AaJa))

for all a G Zn \{0} , m G N and / satisfying (3.8).


498 MAREK NAWROCKI

Observe that in order to prove that t is continuous, it is enough to show

that each norm || • || , p G N, (see (3.0)) is bounded on the T-image of some

neighborhood of zero in Ft(U").

Fix p G N, choose / G N so large that /12',/2~//("+l, < 2"", and take

m := p + n + f . Then, by (3.9) and Lemma 3.2(b), we obtain

\f{a)\ < expiAaJa)) < exp(2_//a^a)) = exp(^(a))

for all a G Z"\{0} and / belonging to the neighborhood of zero V = {/ G

F.(U"): |/(z)| < exp(2-'"n;=1(l - |z,-l)~') for all z e U"} in FfV). Obvi-ously, || • || is bounded on t(F).

4. Multipliers of NfV")

Recall that the Hardy space H (Vn), 0 < p < oo, is the subspace of H(Vn)

consisting of all functions / for which

\H = sup / \f(rœ)\" dm (to) < oo,' 0<r<\ Jt"

while H^CU") is the space of all bounded analytic functions on U".

A family k = {k(a)}nez„ of complex numbers is a multiplier of AJU")

into Hp(V") if for each fi(z) = Ea/(«)^" e NfV") the function lfi(z) =

J2„Ha)fiia)z" belongs to H_(U"). It is well known that for each multiplier l

the induced linear operator / —> À/ is continuous.

In this section, we describe all multipliers of At(U") into the Hardy spaces

HpiV"), 0 < p < oo . For the proof of our main result (Theorem 4.5), we need a

few technical lemmas. We use notation concerning the space M[ß] introduced

in §2.

Lemma 4.1 [5, Chapter 4, §6].

/Jt1 - rto\ dm{to) = 0((1 - r) ) as r —* 1

Definition 4.2. For each k G {1,...,«} , r = (r ) el , c > 0, we define

/, rf(z) = cexp|

Lemma 4.3.

hmsupll/, rJ = 0.c-»0+fc,r

Proof. In the proof we follow [3]. Fix an £ > 0 . Using Lemma 4.1 we can find

a K > 0 such that

, A-

(4.1) / Y\\l - rj\\l - rjtof2 dm fto) < K < oo,Jt" =1



for all reí and k G {I, ... , n}. Choose a subarc / of T with the midpoint

1 so small that mn(Jn)lo%4 < e/3, where Jn := {to = (to.): toj e / for some

j = 1, ... , n}. Using (4.1) and the inequality log(l +cx) < log(l + c) + log2 +

log* (c> 0, x > 1) with x = exp(cY[j=x |1 - r.||l - r.ty .|~ ), we obtain

log( 1 + \fk,rJ) dmn < mfJn)log(l + c) + mfjf) log2 + cK.LChoose cQ > 0 so small that the right side of the above inequality is not greater

than 2e/3 for all 0 < c < c0 . It is easy to see that

lim sup{|/;r c(of\:toGTn\Jn, r Glk , k = I, ... , n} = 0.c—>0+ ■"■>'>'

Therefore, there is 0 < c, < c0 such that /T„,y log(l + \f¿ r c\)dmn < e/3 for

all k , r and 0 < c < c, . Finally, \\fik r c\\ < e for all k, r, and 0 < c < cx .

The proof is complete.

Lemma 4.4. For every e > 0, there is a pGN such that

(4.2) inf sup{|/(a)|: fiG NfV"), \\f\\ < e}/? (a) > 0.<t€Z" ^

Proof. Fix e > 0. There is 0 < d0 < 1 such that sup^. r\\fk r d\\ < e for

all 0 < d < dQ, where fik r d is the function described in Definition 4.2 (see

Lemma 4.3). Take m e N so large that d := 4"/2m < d0. For each permutation

p of {1,...,«}, the operator /(z,, z2, ... , zf -> /(zp(1), zp(2),... , zp(n))

is an isometric automorphism of Nf\]n). Moreover, b Ja) = b (aop) for each

a G Z" . Therefore, to prove (4.2), it suffices to consider only those a G Z" \{0}

that are nonincreasing. For each a of this type, we define

gfz):=exp ¡d\ Z G U ,

where k := Lmia), r. := l-x¡, x¡ := am{a)/ai for /' = 1, 2,

that by equation (3.4), 0 < x¡ < 1/2). Then we have

A^di A^) = E7rIl[(1-''«)y(1-^r2y']

i—r, J ' ;_ i7=0 1=1

— d' —

j=\ J ,=i

x,^l2j + ßi-

/3,=0

w00 A1

7=1J\ e n

(/*,.h

3 dJ k

X]

Tlk 1 = 1 L

V + ßi-\ß,

k (observe

-' + E \Y.jTl/îez* 17=1 í=i

!l2J + !'-l)r{>ßi

M«


500 MAREK NAWROCKI

vk ___^ n,kfor all z e U" (here we identify Z* with Z+ x {0} x • • • x {0} c Z" ). Therefore,

un'= \S«i(*x , ... ,ak,0,... ,0)\

D k r

4nK2;+r')4for all / e N. It is obvious that

2j + a. - 1

ft.>Q<

27-1

(27)! ■

Thus,

w. >d1

¿Wrt$-X\ifi)\iVW t.

for all j G N. Let us set y = yn =the integer part of am(a). Then, using

Stirling's formula we obtain

log«Q > jlogd - jlogj +j- 0(log/) - k(2jlog2j - Ij + <9(log2;)}

k

+ Ylu lo%x¡+ ai lo%ri+j lo&a¡ - 1osq,]1=1

2k+l+ log4kj2k ïïflxi«i)

1=1 /.

+ ^a(.logr(. - 0(log(a, ■■■ak))

¡=\

as (a oo. However,

4k i2kH J «=i

-riv*;>4*K(a)PïïK(«)r(«,-^/2 )2 =

rf2"= 1.

Moreover,

* k k

Y(*i]°Zri = £ «,. log( 1 ~x<) - ~251a«*« = -2^„»-(=i «=i «=i

Consequently,

log«, > i2k + l)iaja) - 1) - 2kaja) - 0(logaJa))

> aja) - 0(logaJa)).

This implies that there are y , «5 > 0 such that

|g(t(«,, ... ,aLJ(i),0, ... , 0)| >áexp(y2~mam(a)) > ôexp(ybja))

for all a e Z"\{0} with a, > ••• > a„ . Now, for the function hjz) :=

dgjz)z'kf; ■ ■ ■ z'f (where k := LJa)), we have ||AJ| = \\dgj = \\fk<rJ < e ,and

h (a) > dSexp(yb (a)) for all a.



Lemma 3.2(c) implies that there exists p e N such that b (a) < ybm(a) for all

a . Finally, ha(a) > dS exp(b (a)) for all a . The proof is finished.

Theorem 4.5. A complex family k = {k(a)}ner is a multiplier of NfV") into

Hp(D"), 0 < p < oo, if and only if

sup ' . ' < oo for some p gN.a ßfa)

Proof. Throughout the proof the letter C will denote many various positive

constants, which are always independent of any individual analytic function or

multi-index.

Suppose that A is a multiplier of Nt(U") into H (Vn). We may assume

0 < p < 1 . The operator A associated with A is continuous, so there is an

e > 0 such that

(4.2) \\lf\\H <l for all fi G NfV"), ||/|| < e.

However, for all g G H (Vn) we have

/

(4.3) \g(a)\ < C

(see [6, Theorem 5]). Thus,

n«;i/p-i■ i=i for all a G Z

(4.4) \k(a)fia)\ < C n l/p-\

J

for all a G Z" and / e /V.(U"), ||/|| < e. By Lemma 4.4, there are ô > 0,

m G N, and a family {fin}aer+ C NfVn) such that ||/J| < e and |/»| >

ô exp(b (a)) for all a . Finally, using Lemma 3.2, we obtain

\k(a)\ < C

(n

YI'i=\

Ai/p-i exp(-bja)) < C(a:f[/p-l) exp(-bja))

^ /^ /i-fw+l), • ,»m+l,l/(«+l), / l / w<Cexp(2 \ax/2 )n ')exp(-bja))

< Cexp(bm+X(a) - bja)) < Cexp((2~l/2 - l)bja))

<Cexp(-bm+fa)) = Cßm+fa)

for all a G if .


502 MAREK NAWROCKI

Suppose now that A = {X(a)} satisfies |A(a)| < C • ß (a) for all a G Z" .

Proposition 3.3 shows that |/(q)| < C/ß x(a) for all a and all / belonging

to some neighborhood of zero V in A^U"). Consequently, £„ |A(a)/(a)| <

CJ2!tß,Aa)/ßM+\ia) < °° (see Lemma 3.2(d)), and so the function Xfi(z) =

Yln^ia)fia)z" *s analytic on U" and continuous on U" for every / e V,

hence 1(V) c H^V"). This completes the proof since neighborhoods of zero

are absorbing.

Remark 4.6. In the case n = 1 , it is easily seen that for each m G N the

sequence {ßm(k)}kez is equivalent to the sequence {exp{-ckl/ )}kez for

some c > 0. Thus, Yanagihara's description of multipliers of Nt into H ,

0 < p < oo , (cf. [17, Theorem 2]) is a particular case of Theorem 4.5.

5. Linear functionals and the Fréchet envelope

Theorem 5.1. To each continuous linear functional T on At(U/!), there corre-

sponds a unique complex family À such that

(5.1) Tf = Yfi<*)KoA) for every fi e 7Y.(U")a

and

(5.2) sup - . | < oo for some pGN.o ßfa)

Conversely, for each sequence {A(a)} satisfying (5.2) .formula (5.1) defines

a continuous linear functional on NfDn).

Proof. The second part of the proof of Theorem 4.5 shows that if À satisfies

(5.2), then the series (5.1) defining Tfi is absolutely convergent for all / and

T is bounded on some neighborhood of zero in Nfif"). Hence, T is well

defined and continuous.

Now suppose that T is a continuous linear functional on Nt{V") and let

/L(«) = Tz" , a G Z" . Then,

(5.3) Tf= lim Tf= lim V f{a)T{zn)r" = lim V/(a)A(a)r0r—>\- r—>\-¿—' r—» 1 - *—'

for ail / e A„(U"), where reí and r" = r"'+' +"n . As has already been

noted, if we show that A satisfies (5.2), it will follow that the series J2fia)Ha)

is absolutely convergent. Consequently, the limits in (5.3) will be equal to the

sum of this series (i.e., (5.1) will hold).

By Theorem 4.5, for the proof of (5.2) it is enough to show that X is a

multiplier of At(U") into HJfif).

For each Ç, zeü" and / e NJV") we define fifz) = fi{Çz). It is easily

seen that ||/J < ||/|| for each fe /V,(U"), so the set {/.:(gU"} is bounded


linear functionals and the fréchet envelope 503

in NfVn). Moreover,

If iQ = YMa)fiia)C = T fe/(a)fV J = 7Y/C) ,

so Xfi G H^iV") for each / e NfUn). In other words, A is a multiplier of

/V,(U") into H^'V").

Let us recall that if X = {X, x) is an F-space whose topological dual X'

separates the points of X , then its Fréchet envelope X is defined to be the com-

pletion of the space iX, xc), where f is the strongest locally convex topology

on X that is weaker than t . In fact, it is known that f is equal to the Mackey

topology of the dual pair iX, X1). See[13]. For each metrizable locally convex

topology £ on X , iX, Ç) is a Mackey space, i.e., £ coincides with the Mackey

topology of the dual pair (X, X'f) (cf. [12, Chapter IV, 3.4]), so the Fréchet

envelope X of X is up to an isomorphism uniquely defined by the following

conditions:

(FE1) X is a Fréchet space,

(FE2) there exists a continuous embedding j of X onto a dense subspace of

X,

(FE3) the mapping y >-* y o j is a linear isomorphism of X' onto X'.

Theorem 5.2. (a) M[ß] is the Fréchet envelope of NfU").

(b) The operator fi ^* {fia)} is an isomorphism ofi FfVn) onto M[ß].

Proof. Let i be the inclusion mapping of NfVn) into FfV") (see Proposition

2.1) and let ea be the a th unit vector in M[ß], i.e., ea(a) = 1 and en(ß) = 0 if

a f ß . It is well known that {en}it&z» is a basis of the nuclear space M[ß]. Let

j = xoi. Then j(za) = ea for each a e Z" , so j is a continuous embedding

of NfV") into M[ß] and j(NfVn)) is dense in M[ß] (see Proposition 3.3).

Condition (FE3) is satisfied because of Proposition 3.1 and Theorem 5.1. Thus,

M[ß] is the Fréchet envelope of Nt(\j"). Obviously, the projective topology

induced on A^U") by j coincides with the projective topology induced by i.

Therefore, t is an isomorphism of Ft(VH) onto M[ß] because of the density

of NfU") in FfU").

The preceding description of the Fréchet envelope of Nt(U") has a nice

application in the proof of the following:

Theorem 5.3. If n > 1 then At(U") is not isomorphic to any complemented

subspace of At(U).

Before proving the theorem, let us recall that a nondecreasing sequence y =

f/j} is said to be a nuclear exponential sequence of finite type if Iim(logy')/v. =

0. For each such sequence y , the finite type power series space A,(y) is defined

to be the space of all complex sequences x = {xA such that

qk(x) = sup \Xj\exp(-y./k) < oo for each k e N.


504 MAREK NAWROCKI

See [4]. We say that y is stable if sup(y-lj/yj) < oo.

Observe that if n = 1 , then the Köthe matrix ß = ß{l) = [ßm(j): j G Z+],

which defines the Fréchet envelope M[ß{l]] of Nt(V), is very simple. Indeed,

ßm(j) = exp(-2~ m' j ' ) (see equation (3.7)) and {j]^} is a stable nuclear

exponential sequence, so we have

Lemma 5.4 [16, Theorem 1]. The Fréchet envelope M[ß([)] of AJU) is iso-

morphic to A.({j ' }).

Lemma 5.5 [4, Proposition 3], Let ô = {6}} be a stable nuclear exponential

sequence of finite type. Then Ax(y) is isomorphic to a subspace of A, («5) if and

only if sup àfy ■ < oo.

We apply these lemmas to prove

Proposition 5.6. If n > I, then the Fréchet envelope M[ß ] of NfVn) is noti tj

isomorphic to any subspace of Ax({j ' }).

Proof. It is easily seen that the space M[ß '] is isomorphic to a subspace of

M[ß ] when n > 2. Therefore, for the proof of the proposition we can

assume that n = 2 .

Let A = {(a, , af G Z+: a2 < a, < af} and Ak = {a G A: a{a2 = k} for

k g Z (note that the set Ak may be empty). Moreover, let j be a unique

bijection of A onto Z+ such that

(a) max j(Ak) < min j(Af if Ak , A^® and k < 1 ,

(b) if a, a G Ak , a = (/', k/i), a = (/', k/i) and / < i , then j(a) <

Jia).

Define a sequence y = {y A by

yj = (axa2)Xß if j = j(a{, af , y =1,2,....

We will show that y is a nuclear exponential sequence of finite type. Indeed,

it is obvious that y is nondecreasing. Moreover, j(ax , af) < \{(s, t) G Z+: t <

s < t2&st < axa2}\ < (axa2)2 for all (a, , af G z\ . Thus,

log;(a, , af) log(Q[Q2)i/3 -* 0 as j(ax , af) -* oo.

yfax,a2) (axa2,

Now, let E be the closed subspace of M[ß ] spanned by the set of the unit

vectors {en : a G A}. It is easily seen that for each a e A and m G N , m > 3 ,

bm(a) = 2~4m/V, , (see (3.6)). Therefore, E is isomorphic to A,(y). Now, by

Lemma 5.5 and Lemma 5.4, for the proof of the proposition it suffices to show

that sup[/ ¡yf] = oo . This is simple. Taking jk = j(k, k ), k = 1,2,...,



we obtain y. = k and

jk>\{(s,t)GA:f(s,t)<j(k,k2)}\

>\{(s,t)GZ2+\{0}:t<s<t2<k2}\

>-±it2-t)>^2k\t=\

Finally, ff ¡y¡ -* oo . The proof is complete.

Proof of Theorem 5.3. Suppose that there exists a continuous projection P of

At(U) onto its subspace X isomorphic to Nt(Vn), where n > 1 . Then P

remains continuous if we equip the spaces Ai(U) and X with their own Mackey

topologies. This implies that the Mackey topology of X is induced by the

Mackey topology of the whole space Nt(V). Let X be the closure of X in the

Fréchet envelope A, ({j} ' ) of Nt(V). Then X is isomorphic to its Fréchet

envelope, which in turn is isomorphic to M[ß{n)]. However, this is impossible

because of Proposition 5.6.

Problem 5.7. Are there any distinct n, meN suchthat NfVn) is isomorphic

to A%(Um)?

Added in proof. After this paper had been completed the author showed that if

nj ^ m then At(U") is not isomorphic to At(Uw) (see The nonisomorphism

of the Smirnov classes of different balls and polydiscs, Bull. Soc. Math. Belg.

Sér B 41 (1989), 307-315).

References

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506 MAREK NAWROCKI

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Institute of Mathematics, A. Mickiewicz University, 60-769 Poznan, Poland


multipliers, linear functionals and the frЙchet

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