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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
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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
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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} ,
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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
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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).
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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
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LINEAR FUNCTIONALS AND THE FRÉCHET ENVELOPE 499
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«
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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.
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LINEAR FUNCTIONALS AND THE FRÉCHET ENVELOPE 501
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 .
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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
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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.
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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,...,
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LINEAR FUNCTIONALS AND THE FRÉCHET ENVELOPE 505
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).
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506 MAREK NAWROCKI
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Institute of Mathematics, A. Mickiewicz University, 60-769 Poznan, Poland
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