Rev. R. Acad. Cien. Serie A. Mat.
SAM
VOL. 97 (2), 2003, pp. 339–363
An´alisis Matem´atico / Mathematical Analysis
Structure theory of power series spaces of infinite type
D. Vogt
Dedicated to the memory of Klaus Floret
Abstract. The paper gives a complete characterization of the subspaces, quotients and complemented subspaces of a stable power series space of infinite type without the assumption of nuclearity, so extending previous work of M. J. Wagner and the author to the nonnuclear case. Various sufficient conditions for the existence of bases in complemented subspaces of infinite type power series spaces are also extended to the nonnuclear case.
Teor´ıa estructural de espacios de series de potencias de tipo infinito Resumen. Este art´ıculo da una caracterizaci´on completa de los subespacios, cocientes y subespacios complementados de los espacios de series de potencias estables de tipo infinito sin el supuesto de nucle- aridad, extendiendo trabajo previo de M. J. Wagner y el autor al caso no nuclear. Se extienden tambi´en varias condiciones suficientes para la existencia de bases en subespacios complementados de espacios de potencias de tipo infinito al caso no nuclear.
1. Introduction
In the present paper we extend the structure theory of nuclear stable power series spaces of infinite type as developed by M. J. Wagner and the author in [22],[29], [30] to the nonnuclear case, i.e. we characterize the subspaces, quotient spaces and complemented subspaces of the Fr´echet space
Λ∞(α) ={x∈KN0 : |x|2t =
∞
X
n=0
|xn|2e2tαn<+∞for allt∈R}.
Hereα= (α0, α1, . . .)is an increasing sequence of nonnegative numbers tending to infinity and satisfying lim sup
n
α2n αn
<+∞. (1)
Condition (1) is equivalent toΛ∞(α)×Λ∞(α)∼= Λ∞(α)and called stability.
In the above mentioned work of Wagner and the author it was additionally assumed that lim sup
n
logn αn
<+∞.
Presentado por Jos´e Bonet.
Recibido: 7 de Mayo de 2003. Aceptado: 4 de Junio de 2003.
Palabras clave / Keywords: power series space, structure theory, (DN), (Ω),α-nuclear Mathematics Subject Classifications: 46A63, 46A45
c
2003 Real Academia de Ciencias, Espa˜na.
This condition is equivalent to the nuclearity ofΛ∞(α). In the present paper we present the theory without this assumption.
While the characterizing conditions remain the same and also the proof of the necessity of these con- ditions, the proof for the sufficiency is essentially different and provides also a new access for the nuclear case. The difference is the following: in the nuclear case the proof was based of the T. and Y. Komura imbedding theorem, resp. its analogue forα-nuclear spaces due to Ramanujan and Terzio˘glu [17], and on a splitting theorem for exact sequences of Fr´echet spaces. The splitting theorem is also valid in the nonnu- clear case. However we do not have an equivalent to the above mentioned imbedding theorems. Therefore the imbedding and quotient maps will be given by a direct construction.
In a last section we consider results of Vogt [23], Aytuna, Krone and Terzio˘glu [1], [2], Wagner [32], Kondakov [9] and Dubinsky and Vogt [6], [7] on the structure of complemented subspaces ofΛ∞(α)and give proofs in the new framework, without assuming nuclearity. Also here in some parts new methods had to be developed.
The paper is partly based on lectures which the author has given in Wuppertal in 1987/88 but never has published. The author thanks M. J. Wagner for useful conversations.
Throughout the paper we will study Fr´echet-Hilbert spaces, i.e. Fr´echet spaces admitting a fundamental system of seminormsk k0 ≤ k k1 ≤. . .given by semiscalar products,kxk2k =hx, xik. If not otherwise stated a fundamental system of seminorms in a Fr´echet-Hilbert space will always be assumed to be of this form. So the local Banach spacesEkare Hilbert spaces.
We will use common notation for locally convex spaces and Fr´echet spaces in particular also for their local Banach spaces and the linking maps between them. The scalar field is alwaysK, whereKis eitherR orC. For all this we refer to [12].
2. Power series spaces of infinite type and related invariants
In the present paper we will study only power series spaces of infinite type as defined above. We will not assume nuclearity. The spaceΛ∞(α)and likewise the sequenceαwill be called stable ifαsatisfies condition (1). For more information on power series spaces we refer to [12, Section 29] and to [4]. For examples see also [18, Chapter 8].
Throughout the paper we use for the local Banach space ofΛ∞(α)with respect to| |tthe notation Λαt :={x= (x0, x1, . . .) : |x|2t :=
∞
X
j=0
|xj|2e2tαj <+∞}.
Of course, these are Hilbert spaces isomorphic to`2, in particularΛα0 =`2.
An analogous extension of the structure theory to the nonnuclear case for finite type power series spaces has been given in [27]. From this paper we quote the following result [27, Theorem 3.2] which is based on [30, Satz 2.4].
Proposition 1 Ifαis stable then there exists an exact sequence
0 −−−−→ Λ∞(α) −−−−→ Λ∞(α) −−−−→ Λ∞(α)N −−−−→ 0.
To describe the characteristic properties of power series spaces and their subspaces, quotient spaces and complemented subspaces we need two types of invariants. The first one describes the asymptotic behavior of the relative semiaxes of the ellipsoids which form the neighborhoods of zero.
LetXbe a linear space andV ⊂U absolutely convex subsets. We set for linear subspacesF, G⊂X:
δ(V, U;F) = inf{δ >0 : V ⊂δU+F} γ(V, U;G) = inf{γ >0 : V ∩G⊂γU}
and with this we notation
δn(V, U) = inf{δ(V, U;F) : dimF ≤n}
γn(V, U) = inf{γ(V, U;G) : codimG≤n}.
δn(V, U)is called then-th Kolmogoroff diameter,γn(V, U)then-th Gelfand diameter. For the behavior of the diameters see [19] or [4, Section I, 6].
We call U a Hilbert disc if U = {x : kxk ≤ 1} and ||x|| is given by a semiscalar product, i.e.
kxk2=hx, xi. The following is well known (see [19, II,3.(3)]).
Lemma 1 IfV anU are Hilbert discs thenδn(V, U) = γn(V, U)for all nand they coincide with the singular numbers of the canonical mapEV ,→EU.
HereEU andEV denote the respective local Hilbert spaces and for the last assertion we assumed that EV ,→EU is compact, i.e. thatV is precompact with respect toU. The following lemma is immediately clear.
Lemma 2 IfV ⊂U are absolutely convex subsets of the linear spaceX,X0⊂X a linear subspace and q:X−→X/X0the quotient map, thenδn(qV, qU)≤δn(V, U)andγn(V∩X0, U∩X0)≤γn(V, U).
We will use the following property of the diameters (cf. [20]).
Lemma 3 Letk k0 ≤ k k1 ≤ k k2 be seminorms on the linear space X and k k21 ≤ k k0k k2, then γn(U2, U0)≤γn2(U1, U0)for alln∈N0.
PROOF. LetG⊂Xbe a linear subspace withcodimG≤nandU2∩G⊂γU0. From the assumption we derive that(γU0)∩U2⊂√
γU1. Inserting the first inclusion into the second we see thatU2∩G⊂√ γU1. Thereforeγ(U2, U1;G) ≤ √
γ and, in consequence, γ(U2, U1;G)2 ≤ γ(U2, U0;G). Since obviously γ(U2, U0;G)2≤γ(U2, U1;G)2γ(U1, U0;G)2we haveγ(U2, U0;G)≤γ(U1, U0;G)2. Taking infima over Gproves the result.
The following definition goes back to Ramanujan-Terzio˘glu [17] (there and in [30] under the name of ΛN(α)-nuclearity). We use the Kolmogoroff diameters for the definition as it is done in [30]. The difference is that we do not assume nuclearity, i. e. we do not assumelim supnlogαn
n <∞.
Definition 1 A Fr´echet-Hilbert spaceEis calledα-nuclear if for every absolutely convex neighborhood Uof zero and everyt >0there is another such neighborhoodV ⊂U, so that
lim
n etαnδn(V, U) = 0.
Remark 1 In view of Lemma 1 we could have used also the condition limn etαnγn(V, U) = 0.
Clearlyα-nuclearity is invariant under topological linear isomorphisms. As an immediate consequence of Lemma 2 we obtain.
Proposition 2 α-nuclearity is inherited by subspaces and quotient spaces.
Lemma 4 Λ∞(α)isα-nuclear.
PROOF. IfUt={x∈Λ∞(α) : |x|t≤1}ands < tthen it is immediate that Ut⊂span{e0, . . . , en−1}+e(s−t)αnUs
wheree0, e1, . . . are the canonical basis vectors. Thereforeδn(Ut, Us)≤e(s−t)αn.
Remark 2 IfcodimG≤nandUt∩G⊂γUsthen there isx∈span{e0, . . . , en} ∩G,kxkt= 1and we have
γ2≥ |x|2s=
n
X
j=0
|xj|2e2sαj ≥e2(s−t)αn
hencee(s−t)αn≤γn(Ut, Us)which means that we have even
γn(Ut, Us) =δn(Ut, Us) =e(s−t)αn. From Lemma 4 and Proposition 2 we obtain:
Proposition 3 IfEis isomorphic to a subspace or a quotient space ofΛ∞(α)thenEisα-nuclear.
Just for the sake of completeness we notice that every closed subspace or quotient space of a Fr´echet- Hilbert space is again a Fr´echet-Hilbert space.
The second type of invariants are those who describe the interpolational properties of the seminorm system definingE. They have been considered, under different names, by Dragilev, Zaharjuta, Dubinsky, Robinson, Wagner and the author. The importance of the properties (DN) and (Ω) is based on the structure theory of power series spaces as developed in [22],[29],[30] and, in particular, on the (DN)-(Ω)-Splitting Theorem (see Theorem 2 below). For the properties of (DN) and (Ω) see [12, Sections 29, 30, 31].
LetEbe a Fr´echet space,k k0≤ k k1≤. . .a fundamental system of seminorms andk k∗0 ≥ k k∗1≥. . . the dual extended real valued norms inE0, where for any seminormk kwe are using the notation
kyk∗= sup
kxk≤1
|y(x)|
fory∈E0.
Definition 2 Ehas property (DN) if there ispso that for anykthere isKandC >0with k k2k≤Ck kpk kK.
k kpis called a dominating norm.
An equivalent formulation is given in the following lemma for the proof of which we refer to [12].
Lemma 5 Ehas property (DN) if and only if there isp, so that for everykand0< τ <1there existsK andC >0with
k kk≤Ck k1−τp k kτK.
Definition 3 Ehas property(Ω)if for everypthere isqsuch that for anykthere is0< ϑ <1andC >0 with
k k∗q ≤Ck k∗p1−ϑk k∗kϑ.
The most prominent example of a space with properties (DN) and(Ω)is the following.
Lemma 6 Λ∞(α)has properties (DN) and(Ω).
PROOF. By H¨older’s inequality we obtain easily fort0< t1< t2
|x|t1≤ |x|
t2−t1 t2−t0
t0 |x|
t1−t0 t2−t0
t2 . (2)
Taking into account that
|y|∗t2=X
j
|yj|2e−2tαj we obtain in the same way
|y|∗t1≤ |y|∗
t2−t1 t2−t0
t0 |y|∗
t1−t0 t2−t0
t2 . (3)
Directly from the definition we see:
Lemma 7 (1) Property (DN)is inherited by subspaces.
(2) Property(Ω)is inherited by quotient spaces.
From Lemmas 6, 7 and Proposition 3 we conclude:
Proposition 4 (1) IfEis isomorphic to a subspace ofΛ∞(α)then it isα-nuclear and has property (DN).
(2) IfEis isomorphic to a quotient space ofΛ∞(α)then it isα-nuclear and has property(Ω).
Of course, the arguments of the proof of Lemma 6 hold also if instead ofΛ∞(α)we take a space of the following form
Λ∞(α, I) ={x∈KI : |x|2t =X
i∈I
|xi|2e2tαi<+∞for allt∈R}.
Hereα= (αi)i∈Iis a family of nonnegative numbers andIis any index set. In particular we have formulas (2) and (3). For any index setJwe setΣ∞(J) = Λ∞(α, I)whereI=N×J andαn,j =n. Moreover we setΣ∞= Σ∞(N). ThenΣ∞(J)andΣ∞have properties (DN)and(Ω).
At this point we want to extend some results of [25] to the case of Fr´echet-Hilbert spaces.
Theorem 1 LetEbe a Fr´echet-Hilbert space andJ a dense subset ofE, then (1) Ehas property (DN) if and only ifEis isomorphic to a subspace ofΣ∞(J).
(2) Ehas property(Ω)if and only ifEis isomorphic to a quotient space ofΣ∞(J).
In particular we have
Corollary 1 LetEbe a separable Fr´echet-Hilbert space then
(1) Ehas property (DN) if and only ifEis isomorphic to a subspace ofΣ∞. (2) Ehas property(Ω)if and only ifEis isomorphic to a quotient space ofΣ∞.
We can read Corollary 1 also as a characterization of the subspaces and quotient spaces ofΣ∞, which is isomorphic toDL2 (see [21] or [24, Theorem 3.2]).
From Theorem 1 we get immediately
Corollary 2 IfEis a Fr´echet-Hilbert space and has property (DN), then there is a one-parameter family (| |t)t∈R of Hilbert norms on E which generates the topology and so that t 7→ log|x|t is convex and increasing for anyx∈E.
Corollary 3 IfEis a Fr´echet-Hilbert space and has property(Ω) then there is a one-parameter family of extended real valued Hilbert seminorms(| |∗t)t∈Rso that the setsBt = {y ∈ E0 : |y|∗t ≤ 1} are a fundamental system of bounded sets inE0andt7→log|x|∗t is convex and decreasing for eachy∈E0.
The proof of Theorem 1 is exactly the same as the proof of Theorems 2.2 and 3.2 in [25]. One has to replace the spaces`∞(I)and`1(I)by`2(I)and apply the following theorem instead of [25, Lemma 1.3.].
For a proof of this theorem see e.g. [12, Section 30].
Theorem 2 Let0 −→ F −→ G −→ E −→ 0 be an exact sequence of Fr´echet-Hilbert spaces and assume that F has property (Ω) and E has property (DN), then the sequence splits.
An essential ingredient in the proof of Theorem 2 is the following lemma (see [12, Lemma 30.7]) which we will use in the sequel.
Lemma 8 LetE andF be Fr´echet-Hilbert spaces. Assume that E has property (DN) with dominating normk k0and thatF has property (Ω). Letk k0be a continuous seminorm onF andk k1be chosen for k k0according to(Ω). Then for everyϕ∈L(E, F1)andε >0there existψ∈L(E, F0),χ∈L(E, F)so thatsupkxk0≤1kψxk0≤εandıo1◦ϕ=ψ+ıo◦χ.
HereF0andF1are the local Hilbert spaces generated byk k0andk k1, respectively.
We will need the following interpolation result for Hilbert scales. For any two index sets I, J and families(αi)i∈I,(βj)j∈Jof nonnegative real numbers we put
Gt= (
x= (xi)i∈I : |x|2t =X
i∈I
e2αit|xi|2<+∞
)
Ht=
x= (xj)j∈J : |x|2t =X
j∈J
e2βjt|xj|2<+∞
.
These are Hilbert spaces, equipped with their natural scalar products. In the following lemmak ktdenotes the norm of an operatorGt→Ht. The following result is well known, see [11, Theorem 1.11].
Lemma 9 LetT ∈ L(G0, H0)and T G1 ⊂ H1. ThenT Gt ⊂ Ht and kTkt ≤ kTk1−t0 kTkt1 for all t∈[0,1].
We will make immediate use of Lemma 9 to prove the following useful fact.
Lemma 10 If k k is a Hilbert norm on Λ∞(α)and | |0 ≤ k k ≤ C| |τ, C ≥ 1. Then there is an automorphismUofΛ∞(α)so that|U x|0=kxkand
|x|t≤ |U x|t≤C|x|t+τ
for allx∈Λ∞(α),t≥0.
PROOF. We denote byH0 the Hilbert space generated byk kand by(,)its scalar product. For every K > τ we consider the canonical mapıoK: ΛαK ,→H0. It is compact, letsnbe the singular numbers. We set
βn =−1 Klogsn. Then the Schmidt representation takes the form
ıoKx=
∞
X
n=0
e−Kβnhx, eniKfn, (4) where(en)n,(fn)n are orthonormal bases inΛαK andH0, respectively. If we setUt ={x : |x|t ≤ 1}, V ={x : kxk ≤1}thenC1Uτ⊂V ⊂U0leads to
δn(UK, U0)≤δn(UK, V)≤Cδn(UK, Uτ) i.e. (see Remark 2)
e−Kαn≤e−Kβn≤Ce−(K−τ)αn. (5) We put
uKx= ((x, fn))n∈N0. We have
|x|0≤ |uKx|0=kxk ≤C|x|τ
and, by use of (5) for the first inequality and (4) for the equation in the middle row, 1
C|uKx|K−τ = 1 C
∞
X
n=0
e2(K−τ)αn|(x, fn)|2
!12
≤
∞
X
n=0
e2Kβn|(x, fn)|2
!12
=|x|K
≤
∞
X
n=0
e2Kαn|(x, fn)|2
!12
=|uKx|K. By use of Lemma 9, applied touKand its inverse, we obtain
|x|t≤ |uKx|t≤C|x|t+τ for all0≤t≤K−τ.
For every k ∈ Nthe set{uK : K ≥ k+τ} is an equicontinuous subset ofL(Λαk+τ,Λαk). Since Λαt −→ Λαs is compact fort > s the set is relatively compact inL(Λαk+τ+1,Λαk−1) for everyk ∈ N. Therefore we may, by use of a diagonal procedure, find a subsequence uKn, so that uKn converges in L(Λαk+τ+1,Λαk−1)for everyk∈N. Since the same applies to(u−1K )Kwe may choose the subsequence so that also(u−1K
n)nconverges inL(Λαk+1,Λαk−1)for allk∈N, and we set forx∈Λ∞(α):
U x= lim
n→∞uKnx, V x= lim
n→∞u−1K
nx.
and certainlyU, V ∈L(Λ∞(α)). Of course, we first take the limits in the local Banach spaces separately and then see that those results define elementsU x∈Λ∞(α),V x∈Λ∞(α), respectively.
It can easily be seen thatU V =V U = id, henceU is an automorphism. We have
|U x|0= lim
n |Uknx|0=kxk
and we have for anyt >0
|x|t≤lim
n |Uknx|t=|U x|t≤C|x|t+τ. This proves the result.
An endomorphismU ofΛ∞(α)for which there are constantsC >0andτ≥0so that
|U x|t≤C|x|t+τ, x∈Λ∞(α)
for allt >0we call uniformly tame. A description of uniformly tame endomorphisms is contained in the following lemmas.
For any endomorphismU ∈L(Λ∞(α))there is a matrix(uk,j)k,j∈N0so that U x=
∞
X
j=0
uk,jxj
k∈N0
, x∈Λ∞(α).
Of course, we haveuk,j =hU ej, ekiifekandejdenote the canonical basis vectors andh , ithe`2-scalar product.
Lemma 11 Letαbe strictly increasing andU ∈L(Λ∞(α))be uniformly tame, then its matrix is upper triangular.
PROOF. From the continuity estimates we get
|uk,j|etαk≤ |U ej|t≤C|ej|t+τ =C e(t+τ)αj. Therefore
|uk,j| ≤C et(αj−αk)+τ αj for allt >0which implies the result.
Ifαis not necessarily strictly increasing we have to replace upper triangular by blockwise upper trian- gular, where the blocks are given by the sets of indices on whichαis constant.
Lemma 12 Let α be strictly increasing and S ∈ L(`2) have an upper triangular matrix, then S ∈ L(Λ∞(α))andSis uniformly tame withτ = 0.
PROOF. We may assume thatkSkL(`2)= 1. Then in particular|sj,j| ≤1for allj. Let(d0, d1, . . .)with supj|dj| ≤4be a sequence so that|sj,j +dj|= 4for allj.
We setDx= (d0x0, d1x1, . . .)andH =S+D. LetH0x= (ho,ox0, h1,1x1, . . .)be the diagonal part ofH andH+ =H −H0thenH+ =S+ whereS0andS+ are defined in an analogous way. Therefore kH+k=kS+k ≤ kSk+kS0k ≤2, all norms taken inL(`2).
Because ofH = H0(I+H0−1H+)andkH0−1H+k ≤ 12 the operatorH is invertible inL(`2)and kH−1k ≤1. So we have
|x|0≤ |Hx|0≤5|x|0.
We apply Lemma 10 tokxk=|Hx|0and get a uniformly tame automorphismU ofΛ∞(α)with
|x|t≤ |U x|t≤5|x|t, x∈Λ∞(α)
for allt >0, so that|U x|0=|Hx|0for allx. By Lemma 11 the matrix ofU is upper triangular.
ThereforeV = U ◦H−1 is unitary in`2and has an upper triangular matrix. This implies that V is diagonal and|vj,j|= 1for alljand therefore|Hx|t=|V−1U x|t≤5|x|tfor allx∈Λ∞(α)andt >0.
Finally we obtain
|Sx|t≤ |Hx|t+|Dx|t≤9|x|t, x∈Λ∞(α) for allt >0. This proves the result.
Since we can modify anyαto a strictly increasing sequence without changingΛ∞(α)we obtain from Lemma 12 as an immediate consequence:
Lemma 13 LetS∈L(`2)have an upper triangular matrix, thenS∈L(Λ∞(α)).
3. Subspaces of power series spaces of infinite type
We characterize in this section the subspaces of a stable power series spaceΛ∞(α). We assume that that the Fr´echet-Hilbert spaceEhas property (DN) and isα-nuclear. By Corollary 1 the spaceEis isomorphic to a subspace ofΣ∞. We may assume that it is a subspace ofΣ∞.
By| |twe denote the canonical norms onΣ∞and also their restrictions toE.
We assume chosen a sequence0 =t0 < t1 < . . . withlimktk =∞. We setk kk =| |tk onEand Uk={x∈E : kxkk ≤1}.
By definition thek k0≤ k k1≤. . .are Hilbertian norms onEso that with suitable0< τk<1 k kk≤ k k1−τk−1kk kτk+1k . (6) By the choice of our seminorms the spaceEis countably normed, i.e. the canonical mapılk:Ek −→Elis injective for allk≥l.
The fundamental lemma in this section is:
Lemma 14 Assume thatδn(Uk, U0)≤e−kαnfor alln≥n0. Then there is a sequenceβwithβn ≥an for alln≥n0and an injective mapϕk ∈L(E,Λ∞(β)), so that for allx∈E
1. |ϕk(x)|0≤2kxk0, 2. |ϕk(x)|k≥ 12kxkk.
Moreoverϕkextends to an isomorphismϕˆk :E0−→`2= Λβ0.
PROOF. We sets = 2tk,E(s)the local Banach space of| |sandı(s)l : El −→E(s),ıl(s) :E(s)−→ El
the respective linking maps. By assumption the mapı0k, hence alsoı0(s):E(s)−→E0is compact. Letsn
be the singular numbers ofı0(s). We set
βn =−1 2klogsn. Then the Schmidt representation takes the form
ı0(s)x=
∞
X
n=0
e−2kβnhx, enisfn. (7) Here(en)nis an orthonormal basis ofE(s)and(fn)nan orthonormal basis ofE0.
The choice ofsand (2) applied toΣ∞impliesk k2k ≤ k k0| |s. Ifδn(Uk, U0)≤e−kαnwe obtain, by use of Lemmas 1 and 3, withU(s)={x∈E : |x|s≤1}
e−2kβn =δn(U(s), U0)≤δ2n(Uk, U0)≤e−2kαn. Thereforeαn≤βnfor alln≥n0.
We setϕx= (hx, fni0)n∈N0 and obtain a unitary mapϕ:E0−→`2= Λβ0. From (7) we see that ϕ◦ı0(s)(x) = (e−2kβnhx, enis)n
which means thatϕ◦ı0(s)is a unitary mapE(s) −→Λβ2k. We denote byuits inverse. Thenuis unitary Λβ2k−→E(s)and we have
|u ξ|s=|ξ|2k
kı0(s)u ξk0=|ξ|0
for allξ∈Λβ2k. The latter because obviously
ı0(s)◦u=ϕ−1|Λβ 2k
. (8)
By Lemma 9, applied for maps between the Hilbert spaces ofΛβt andΣtwe obtain
kık(s)u ξkk ≤ |ξ|k. (9)
Notice that, by assumption,E ⊂Σ∞and therefore, by natural identification,Ek ⊂Σtk,E(s) ⊂Σs. By this identification we havekı0(s)u ξk0=|uξ|0andkık(s)u ξkk =|uξ|tk.
We apply Lemma 8 (withε=12) toϕ◦ı0(s)∈L(E(s),Λβ2k). We obtain mapsϕk∈L(E,Λ∞(β)), ψ∈ L(E0,Λβk)so thatsupkxk0≤1|ψx|k ≤12 and
ϕ◦ı0=ϕk+ψ◦ı0. We have to verify the desired properties forϕk.
Forx∈Ewe have
|ϕkx|0≤|ϕ(ı0x)|0+|ψ(ı0x)|k
≤2kxk0.
This proves part 1. Moreoverϕk extends toϕˆk ∈L(E0,Λβ0)and we haveϕˆk =ϕ−ψ=ϕ(I−ϕ−1ψ).
Sincekϕ−1ψkL(E0)≤12 the mapϕˆkis invertible.
The mapı0is injective, hence alsoϕk= ˆϕk◦ı0. We may writeϕˆ−1k by its Neumann series
ˆ
ϕ−1k =ϕ−1+ϕ−1ψϕ−1
∞
X
n=0
(ψϕ−1)n
=ϕ−1+ϕ−1◦v where
v=ψ
∞
X
n=0
(ϕ−1ψ)nϕ−1.
Hencev∈L(Λβ0,Λβk)andkvkL(Λβ
0,Λβk)≤1.
Forx∈Eandξ=ϕkxwe obtain
ı0x=ϕ−1(ξ) +ϕ−1(vξ) =ϕ−1(ξ+vξ). (10) Because of (9) the mapık(s)◦uextends touk ∈L(Λβk, Ek). Due to (8), forη∈Λβkwe haveϕ−1η=ı0kukη.
Withη=ξ+vξwe obtain from (10), using thatı0kis injective,ıkx=uk(ξ+vξ).
Therefore we have
kxkk ≤ |ξ+vξ|k ≤2|ξ|k. This completes the proof.
Lemma 15 If lim supn α2n+1α
n < d < +∞ and βn ≥ αn for all n ≥ n0, then there exists a strictly monotonous sequence(n(ν))ν∈N0andb≥0so thatβν ≤αn(ν)≤b+dβνfor allν ∈N0.
PROOF. We setm(ν) = sup{n : αn ≤βν}andn(ν) =m(ν) +ν+ 1. Then by assumption we have m(ν)≥νforν ≥n0andn(ν)is strictly increasing. Moreoverβν < αn(ν)for allν. We choosen1≥n0
so thatα2n+1≤dαnforn≥n1. Forν ≥n1we have
αn(ν)≤α2m(ν)+1≤dαm(ν)≤dβν. Settingb= sup{αn(ν) : ν < n1}we obtain the result.
Lemma 16 Under the assumptions of Lemma 14 and the assumption thatlim supn αα2n
n <+∞we obtain:
for everykthere is an injective mapϕk∈L(E,Λ∞(α))so that for allx∈E 1. |ϕk(x)|0≤2kxk0
2. |ϕkx|k≥ 12kxkk.
PROOF. We chooseβ andϕk as in Lemma 14. We use Lemma 15 to find a sequencen(ν)withβν ≤ αn(ν)≤b+dβνand define forξ= (ξ0, ξ1, . . .)a sequenceϕ ξby
(ϕ ξ)n =
(ξν ifn=n(ν) 0 otherwise.
Thenϕgives an isomorphic imbeddingΛ∞(β),→ Λ∞(α)and an isometric imbedding`2 ,→ `2. If we replaceϕkbyφk:=ϕ◦ϕk, thenφk ∈L(E,Λ∞(α))is injective, we have
|φk(x)|0≤ |ϕk(x)|0≤2kxk0 and withξ=ϕkx
|φkx|k =|ϕ(ϕkx)|k= X
ν
|ξν|2e2kαn(ν)
!12
≥ X
ν
|ξν|2e2kβν
!12
=|ϕkx|k≥ 1
2kxkk.
We are now in the position to prove the main theorem of this section. For the nuclear case see [30, Satz 3.2].
Theorem 3 Letαbe stable. A Fr´echet spaceEis isomorphic to a subspace ofΛ∞(α)if and only ifEis anα-nuclear Fr´echet-Hilbert space with property (DN).
PROOF. IfEis isomorphic to a subspace ofΛ∞(α)then it is Fr´echet-Hilbert and, by Proposition 4, it is α-nuclear and has property (DN).
We have to show the converse implication. We setn(i, k) = 2k+i2k+1−1and get a bijective map N0×N0−→N0. We setαi,k :=αn(i,k)and obtain an isomorphismΛ∞(α)∼= Λwhere
Λ :={x= (xi,k)i,k∈N0 : kxk2t =X
i,k
|xi,k|2e2tαi,k <+∞for allt∈R}.
The isomorphism is given by(xn)n∈N0 7→(xn(i,k))i,k∈N0.
In E we choose a fundamental system of Hilbertian norms as in Lemma 14. For everyk ∈ N0 we chooseϕk∈L(E,Λ∞(α))according to Lemma 16 and set
ϕk(x) = (ϕi,k(x))i∈N0.
We define forx∈E
φ(x) := (2−ke−kαi,kϕi,k(x))i,k∈N0. We have to show thatφ(x)∈Λandφ∈L(E,Λ).
From the stability ofαwe getbk ≥0,dk ∈Nso that
αi≤αi,k≤bk+dkαi.
We may assume thatbk≤bk+1anddk≤dk+1for allk. Now, we fixmand chooseM, Csuch that
|ϕkx|mdm ≤CkxkM
fork= 0, . . . , m−1. We get kφxk2m≤
m−1
X
k=0
4−k
∞
X
i=0
|ϕi,k(x)|2e2(m−k)αi,k
! +
∞
X
k=m
4−k
∞
X
i=0
|ϕi,k(x)|2
!
≤
m−1
X
k=0
4−ke2mbm|ϕk(x)|2md
m+
∞
X
k=m
4−k|ϕk(x)|20
≤4
3(e2mbmC2+ 4)kxk2M. On the other hand we have form∈N
kφxk2m≥2−m
∞
X
i=0
|ϕi,m(x)|2e2mαi,m
!12
≥2−m|ϕm(x)|m≥2−m−1kxkm.
Thereforeφis an isomorphic imbeddingE ,→Λ∼= Λ∞(α).
4. Quotient spaces of power series spaces of infinite type
In this section we characterize the quotient spaces of a stable power series space Λ∞(α). We assume that that the Fr´echet-Hilbert spaceE has property(Ω)and isα-nuclear. By Corollary 1 the spaceE is isomorphic to a quotient ofΣ∞. Letq: Σ∞−→Ebe the quotient map.
By| |twe denote the quotient seminorms underqof the canonical norms| |tonΣ∞.
We assume chosen a sequence 0 = t0 < t1 < . . . with limktk = ∞. We set k kk = | |tkand Uk={x : kxkk ≤1}.
By definition thek k0≤ k k1≤. . .are Hilbertian seminorms onEso that with suitable0 < ϑk <1 andCk>0
k k∗k≤Ckk k∗k−11−ϑkk k∗k+1ϑk. (11) The fundamental lemma in this section is:
Lemma 17 Assume that δn(Uk+1, Uk) ≤ e−2pαn for all n ≥ n0. Then there is a sequence β with βn≥αnfor alln≥n0and a mapϕk ∈L(Λ∞(β), E)so that for allξ∈Λ∞(β)
1. kϕkξkk ≤2|ξ|0
2. kϕ ξkk+1≥ 12|ξ|p.
Moreoverϕkextends to an isomorphismϕˆk: `2= Λβ0 −→Ek.
PROOF. By assumption the mapıkk+1is compact. We set βn=−1
2plogδn(Uk+1, Uk).
Then the Schmidt representation takes the form ıkk+1x=
∞
X
n=0
e−2pβnhx, enik+1fn
where(en)n∈N0is an orthonormal sequence inEk+1,(fn)n∈N0an orthonormal basis ofEk. As in the proof of Lemma 14, we getαn≤βnfor alln≥n0. We set
ϕξ=
∞
X
n=0
ξne2pβnen
and obtain an isometryϕfromΛβ2pintoEk+1. Because of ıkk+1ϕξ=
∞
X
n=0
ξnfn
the mapıkk+1◦ϕextends to a unitary mapχ: `2= Λβ0 −→Ek.
We puts = 12(tk+1−tk). We denote byE(s) the local Hilbert space of| |sand byı(s), ı(s)k+1, ık(s) the respective canonical maps. We apply Lemma 8 to the map ϕ ∈ L(Λ∞(β), Ek+1) defined by ϕ and the seminorms| |s ≤ k kk+1 onE. We obtainϕk ∈ L(Λ∞(β), E), ψ ∈ L(Λ∞(β), E(s))so that sup|ξ|0≤1|ψξ|s≤12 and
ı(s)k+1◦ϕ=ı(s)◦ϕk+ψ.
We have to verify the desired properties forϕk. We have
ık◦ϕk=ıkk+1◦ϕ−ık(s)◦ψ.
Thereforekϕkξkk ≤2|ξ|0forξ∈E. Moreoverık◦ϕk extends to a mapϕˆk ∈L(`2, Ek)which can be written as
ˆ
ϕk=χ◦(id−χ−1◦ık(s)◦ψ).ˆ
Hereψˆdenotes, as always, the continuous extension. Sinceχis invertible andkχ−1◦ık(s)◦ψkˆ L(`2)≤ 12 the mapϕˆkis invertible.
We have forx∈Ek
χ−1(x) = (hx, fnik)n∈N0. Therefore forx∈Ek+1
χ−1(ıkk+1x) = (e−2pβnhx, enik+1)n∈N0. Forx∈Ewe defineh(x) :=χ−1(ıkx)and obtain
|h(x)|0≤kxkk
|h(x)|2p≤kxkk+1. Therefore by Lemma 9, applied toΣ∞andΛ∞(β), we get
|h(x)|p≤|x|s.
This implies thatχ−1◦ık(s)◦ψextends to a mapT ∈ L(Λβ0,Λβp)and we havekTkL(Λβ
0,Λβp) ≤ 12. Thereforeid−Tis invertible inL(Λβp), withk(id−T)−1kL(Λβ
p)≤2.
Forξ∈Λ∞(β)we setx= (id−T)ξand haveϕk(ξ) =χ(x)hencex=h(ϕk(ξ))which implies
|ξ|p≤2|x|p≤2kϕk(ξ)kk+1. This completes the proof.
We assume now thatαis stable and may choosed∈N, so that lim sup
n
α2n+1 αn
< d.
Lemma 18 Under the assumptions of Lemma 17 and the assumption thatδn(Uk+1, Uk) ≤e−2dpαn for n≥n0andlim supnα2n+1α
n < dwe obtain: there is a mapϕk∈L(Λ∞(α), E)andck >0, so that 1. kϕkξkk ≤2|ξ|0for allξ∈Λ∞(α)
2. |ϕ0ky|∗p≥ckkyk∗k+1for ally∈E0.
Moreoverϕkextends to a surjectionϕˆk: Λα0 −→Ek.
PROOF. From Lemma 17, applied to(dαn)n, we get for givenka sequenceβwithβn≥dαnforn≥n0
and a mapϕk∈L(Λ∞(β), E)with the properties1.and2.in Lemma 17.
From Lemma 15 we get a increasing sequence(n(ν))ν∈N0 andb >0so that 1
dβν≤αn(ν)≤b+βν. We set, as in the proof of Lemma 16
(ϕξ)n=
(ξν ifn=n(ν) 0 otherwise.
Thenϕgives an isomorphic imbeddingΛ∞(β),→Λ∞(α)and an isometric imbedding`2,→`2. We have
|ϕx|t≤etb|x|tfor allt∈R.
We defineL∈L(Λ∞(α),Λ∞(β))by
(Lξ)ν∈N0 = (ξn(ν))ν∈N0.
ThenL◦ϕ= idand|Lξ|t≤ |x|dtfor allt≥0. We putφk=ϕk◦L. Thenφk∈L(Λ∞(α), E).
Moreover1.is satisfied since|Lξ|0 ≤ |ξ|0and, becauseLis surjective, alsoφˆkis surjective. As for2.
we notice that fory∈E0 1
2kyk∗k+1≤|ϕ0ky|∗p=|(ϕ0◦L0◦ϕ0k)y|∗p
≤epb|(L0◦ϕ0k)y|∗p=epb|φ0ky|∗p.
We are now ready to prove the main theorem of this section. For the nuclear case see [30, Satz 3.4].
Theorem 4 Letαbe stable. A Fr´echet spaceEis isomorphic to a quotient space ofΛ∞(α)if and only if Eis anα-nuclear Fr´echet-Hilbert space with property(Ω).
PROOF. IfEis isomorphic to a quotient space ofΛ∞(α)then it is Fr´echet-Hilbert and, by Proposition 4, it isα-nuclear and has property(Ω).
We have to show the converse implication. As in the proof of Theorem 3 we setn(i, k) = 2k+i2k+1−1 andαi,k:=αn(i,k)and use the isomorphic representationΛ∞(α)∼= Λwhere
Λ :={x= (xi,k)i,k∈N0 : kxk2t =X
i,k
|xi,k|2e2tαi,k <+∞fore allt∈R}.
For everyk ∈ N0 we choosedk ∈N,bk ≥ 0so thatαi,k ≤bk+dkαi for alliand setpk = kdk. We choose by induction a sequencet0 = 0< t1 < t2 < . . . so that the seminormsk kk =| |tk satisfy the assumptions of Lemma 18 withp=pk. For eachk∈N0letψkbe the map from Lemma 18.
Forx= (xi,k)i,k∈N0 andk∈N0setxk = (xi,k)i∈N0. Sincei≤n(i, k), henceαi ≤αn(i,k)=αi,k, we havexk∈Λ∞(α)and
∞
X
k=0
|xk|2t ≤
∞
X
k=0
∞
X
i=0
|xi,k|2e2tαi,k =kxk2t for allt≥0.
We put
Ψ(x) :=
∞
X
k=1
2−k2ψk(xk)
forx= (xk)k= (xi,k)i,k ∈Λ. We show that the series converges and definesΨ∈L(Λ, E).
This follows from
kΨxk2m≤
m−1
X
k=1
2−k2kψkxkkm+
∞
X
k=m
2−k2kψkxkkk
!2
≤C2
m−1
X
k=1
|xk|2M+ 4
∞
X
k=m
|xk|20
≤(C2+ 4)
∞
X
k=0
|xk|2M
≤(C2+ 4)kxk2M with suitableCandM depending fromm.
Sinceψk ∈L(Λ∞(α), E)andψˆk is surjective from`2ontoEk, we see thatR(Ψ)is dense inE. Let mbe given. We obtain fory∈E0:
kΨ0yk∗m≥2−m2 sup{|(ψ0m(y))(ξ)| : X
i
|ξi|2e2mαi,m≤1}
≥2−m2 sup{|(ψ0m(y))(ξ)| : |ξ|mdm ≤e−mbm}
≥2−m2 ·e−mbm|ψm0 (y)|∗mdm
≥2−m2e−mbmcmdmkyk∗m+1.
The surjectivity criterion [12, 26.1] then shows thatΨis surjective.
5. Complemented subspaces of power series spaces of infi- nite type
Our next task is to characterize the complemented subspaces ofΛ∞(α)by invariants and study their struc- ture.
First we prove a partial replacement for the Ramanujan and Terzio˘glu imbedding theorem from [17].
Lemma 19 IfQis a quotient ofΛ∞(α), equipped with the quotient seminorms k ktof the norms| |t, t ∈R, then for anykthere isSk ∈L(Q,Λ∞(α)), so thatsupkxk
k≤1|Skx|0 <+∞andSk induces an isometrySˆk:Qk,→Λ00=`2.
PROOF. We may assumek = 0anddimQ0 = ∞. Letϕ: Λ∞(α) −→ Qbe the quotient map. For K >0the mapı0K:QK −→Q0is compact. Letsnbe its singular numbers. We set
βn =−1 Klogsn. Then its Schmidt representation takes the form
ı0Kx=X
n
e−Kβnhx, eniKfn,
where(en)n∈N0is an orthonormal system inQK,(fn)n∈N0an orthonormal basis ofQ0. By Remark 2 and Lemma 2 we get
e−Kβn≤e−Kαn i.e.βn≥αnfor alln.
We set forx∈Λ∞(α)
TKx:= (hx, fni)n. Then we have
|TKx|β0 =kxk0, |TKx|βK=kxkK, hence
|TKx|α0 =kxk0, |TKx|αK≤kxkK. and therefore by Lemma 9, applied toTK◦q,
|Tkx|αk ≤ kxkkfor all0≤k≤K.
Here| |βt and| |αt denote the norms inΛ∞(β)andΛ∞(α), respectively.
We argue like in the proof of Lemma 10 to find a subsequence (TKn)n∈N and an operator T ∈ L(Q,Λ∞(α))so thatTKnx−→T xfor allx∈Q.
For everyx∈Qwe have
|T x|0= lim
n |TKnx|0=kxk0. SettingS0:=Twe obtain the result.
As an immediate consequence we obtain:
Lemma 20 IfQis a quotient ofΛ∞(α)then there is an imbeddingS:Q ,→Λ∞(α)N. PROOF. We setSx= (Skx)k∈N,Skas in Lemma 19.
We can now give the characterization of the complemented subspaces ofΛ∞(α). For the nuclear case see [30, Satz 3.5].
Theorem 5 Letαbe stable. A Fr´echet spaceE is isomorphic to a complemented subspace ofΛ∞(α)if and only ifEis anα-nuclear Fr´echet-Hilbert space with properties (DN) and(Ω).
