Structure theory of power series spaces of infinite type

RACSAM Rev. R. Acad. Cien. Serie A. Mat. 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 nuclearidad, 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

|xn |2 e2tαn < +∞ for all t ∈ R}.

n=0

Here α = (α0 , α1 , . . .) is an increasing sequence of nonnegative numbers tending to infinity and satisfying lim sup n

α2n < +∞. αn

(1)

Condition (1) is equivalent to Λ∞ (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

log n < +∞. 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.

339

D. Vogt

This condition is equivalent to the nuclearity of Λ∞ (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 conditions, 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 for α-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 nonnuclear 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 of Λ∞ (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 seminorms k 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 spaces Ek are 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 always K, where K is either R or C. 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 space Λ∞ (α) and likewise the sequence α will be called stable if α satisfies 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 of Λ∞ (α) with respect to | |t the notation 2 Λα t := {x = (x0 , x1 , . . .) : |x|t :=

∞ X

|xj |2 e2tαj < +∞}.

j=0

Of course, these are Hilbert spaces isomorphic to `2 , in particular Λα 0 = `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 If α is 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. Let X be a linear space and V ⊂ U absolutely convex subsets. We set for linear subspaces F, G ⊂ X: δ(V, U ; F ) = inf{δ > 0 : V ⊂ δU + F } γ(V, U ; G) = inf{γ > 0 : V ∩ G ⊂ γU } 340

Power series spaces of infinite type

and with this we notation δn (V, U ) = inf{δ(V, U ; F ) : dim F ≤ n} γn (V, U ) = inf{γ(V, U ; G) : codim G ≤ n}. δn (V, U ) is called the n-th Kolmogoroff diameter, γn (V, U ) the n-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 If V an U are Hilbert discs then δn (V, U ) = γn (V, U ) for all n and they coincide with the singular numbers of the canonical map EV ,→ EU .  Here EU and EV denote the respective local Hilbert spaces and for the last assertion we assumed that EV ,→ EU is compact, i.e. that V is precompact with respect to U . The following lemma is immediately clear. Lemma 2 If V ⊂ U are absolutely convex subsets of the linear space X, X0 ⊂ X a linear subspace and q : X −→ X/X0 the quotient map, then δn (qV, qU ) ≤ δn (V, U ) and γn (V ∩X0 , U ∩X0 ) ≤ γn (V, U ).  We will use the following property of the diameters (cf. [20]). Lemma 3 Let k k0 ≤ k k1 ≤ k k2 be seminorms on the linear space X and k k21 ≤ k k0 k k2 , then γn (U2 , U0 ) ≤ γn2 (U1 , U0 ) for all n ∈ N0 . P ROOF. Let G ⊂ X be a linear subspace with codim G ≤ n and U2 ∩ G ⊂ γU0 . From the assumption we √ √ derive that (γU0 ) ∩ U2 ⊂ γU1 . Inserting the first inclusion into the second we see that U2 ∩ G ⊂ γU1 . √ Therefore γ(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)2 γ(U1 , U0 ; G)2 we have γ(U2 , U0 ; G) ≤ γ(U1 , U0 ; G)2 . Taking infima over G proves 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 n is that we do not assume nuclearity, i. e. we do not assume lim supn log αn < ∞. Definition 1 A Fr´echet-Hilbert space E is called α-nuclear if for every absolutely convex neighborhood U of zero and every t > 0 there is another such neighborhood V ⊂ U , so that lim etαn δn (V, U ) = 0. n

Remark 1 In view of Lemma 1 we could have used also the condition lim etαn γn (V, U ) = 0.  n

Clearly α-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) is α-nuclear. 341

D. Vogt

P ROOF.

If Ut = {x ∈ Λ∞ (α) : |x|t ≤ 1} and s < t then it is immediate that Ut ⊂ span{e0 , . . . , en−1 } + e(s−t)αn Us

where e0 , e1 , . . . are the canonical basis vectors. Therefore δn (Ut , Us ) ≤ e(s−t)αn .



Remark 2 If codim G ≤ n and Ut ∩ G ⊂ γUs then there is x ∈ span{e0 , . . . , en } ∩ G, kxkt = 1 and we have n X γ 2 ≥ |x|2s = |xj |2 e2sαj ≥ e2(s−t)αn j=0

hence e(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 If E is isomorphic to a subspace or a quotient space of Λ∞ (α) then E is α-nuclear.  Just for the sake of completeness we notice that every closed subspace or quotient space of a Fr´echetHilbert space is again a Fr´echet-Hilbert space. The second type of invariants are those who describe the interpolational properties of the seminorm system defining E. 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]. Let E be a Fr´echet space, k k0 ≤ k k1 ≤ . . . a fundamental system of seminorms and k k∗0 ≥ k k∗1 ≥ . . . the dual extended real valued norms in E 0 , where for any seminorm k k we are using the notation kyk∗ = sup |y(x)| kxk≤1

for y ∈ E 0 . Definition 2 E has property (DN) if there is p so that for any k there is K and C > 0 with k k2k ≤ Ck kp k kK . k kp is called a dominating norm. An equivalent formulation is given in the following lemma for the proof of which we refer to [12]. Lemma 5 E has property (DN) if and only if there is p, so that for every k and 0 < τ < 1 there exists K and C > 0 with k kk ≤ Ck k1−τ k kτK .  p Definition 3 E has property (Ω) if for every p there is q such that for any k there is 0 < ϑ < 1 and C > 0 with 1−ϑ ϑ k k∗q ≤ Ck k∗p k k∗k . The most prominent example of a space with properties (DN) and (Ω) is the following. Lemma 6 Λ∞ (α) has properties (DN) and (Ω). 342

Power series spaces of infinite type

P ROOF.

By H¨older’s inequality we obtain easily for t0 < t1 < t2 t2 −t1

t1 −t0

|x|t1 ≤ |x|tt02 −t0 |x|tt22 −t0 .

(2)

Taking into account that 2

|y|∗t =

X

|yj |2 e−2tαj

j

we obtain in the same way t2 −t1 t −t0

|y|∗t1 ≤ |y|t∗0 2

t1 −t0 t −t0

|y|∗t2 2

. 

(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 (DN).

(1) If E is isomorphic to a subspace of Λ∞ (α) then it is α-nuclear and has property

(2) If E is isomorphic to a quotient space of Λ∞ (α) then it is α-nuclear and has property (Ω).



Of course, the arguments of the proof of Lemma 6 hold also if instead of Λ∞ (α) we take a space of the following form X Λ∞ (α, I) = {x ∈ KI : |x|2t = |xi |2 e2tαi < +∞ for all t ∈ R}. i∈I

Here α = (αi )i∈I is a family of nonnegative numbers and I is any index set. In particular we have formulas (2) and (3). For any index set J we set Σ∞ (J) = Λ∞ (α, I) where I = N × J and αn,j = n. Moreover we set Σ∞ = Σ∞ (N). Then Σ∞ (J) and Σ∞ 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 Let E be a Fr´echet-Hilbert space and J a dense subset of E, then (1) E has property (DN) if and only if E is isomorphic to a subspace of Σ∞ (J). (2) E has property (Ω) if and only if E is isomorphic to a quotient space of Σ∞ (J).  In particular we have Corollary 1 Let E be a separable Fr´echet-Hilbert space then (1) E has property (DN) if and only if E is isomorphic to a subspace of Σ∞ . (2) E has property (Ω) if and only if E is isomorphic to a quotient space of Σ∞ .  We can read Corollary 1 also as a characterization of the subspaces and quotient spaces of Σ∞ , which is isomorphic to DL2 (see [21] or [24, Theorem 3.2]). From Theorem 1 we get immediately 343

D. Vogt

Corollary 2 If E is 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 any x ∈ E.  Corollary 3 If E is a Fr´echet-Hilbert space and has property (Ω) then there is a one-parameter family of extended real valued Hilbert seminorms (| |∗t )t∈R so that the sets Bt = {y ∈ E 0 : |y|∗t ≤ 1} are a fundamental system of bounded sets in E 0 and t 7→ log |x|∗t is convex and decreasing for each y ∈ E 0 .  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 Let 0 −→ 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 Let E and F be Fr´echet-Hilbert spaces. Assume that E has property (DN) with dominating norm k k0 and that F has property (Ω). Let k k0 be a continuous seminorm on F and k k1 be chosen for k k0 according to (Ω). Then for every ϕ ∈ L(E, F1 ) and ε > 0 there exist ψ ∈ L(E, F0 ), χ ∈ L(E, F ) so that supkxk0 ≤1 kψxk0 ≤ ε and ıo1 ◦ ϕ = ψ + ıo ◦ χ.  Here F0 and F1 are the local Hilbert spaces generated by k k0 and k 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∈J of nonnegative real numbers we put ( Gt =

)

x = (xi )i∈I :

|x|2t

=

X

2αi t

e

2

|xi | < +∞

i∈I

    X Ht = x = (xj )j∈J : |x|2t = e2βj t |xj |2 < +∞ .   j∈J

These are Hilbert spaces, equipped with their natural scalar products. In the following lemma k kt denotes the norm of an operator Gt → Ht . The following result is well known, see [11, Theorem 1.11]. t Lemma 9 Let T ∈ L(G0 , H0 ) and T G1 ⊂ H1 . Then T Gt ⊂ Ht and kT kt ≤ kT k1−t 0 kT k1 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 automorphism U of Λ∞ (α) so that |U x|0 = kxk and |x|t ≤ |U x|t ≤ C|x|t+τ for all x ∈ Λ∞ (α), t ≥ 0. 344

Power series spaces of infinite type

P ROOF. We denote by H0 the Hilbert space generated by k k and by ( , ) its scalar product. For every K > τ we consider the canonical map ıoK : Λα K ,→ H0 . It is compact, let sn be the singular numbers. We set 1 βn = − log sn . K Then the Schmidt representation takes the form ∞ X

ıoK x =

e−Kβn hx, en iK fn ,

(4)

n=0

where (en )n , (fn )n are orthonormal bases in Λα K and H0 , respectively. If we set Ut = {x : |x|t ≤ 1}, V = {x : kxk ≤ 1} then C1 Uτ ⊂ V ⊂ U0 leads 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 uK x = ((x, fn ))n∈N0 . We have |x|0 ≤ |uK x|0 = kxk ≤ C|x|τ and, by use of (5) for the first inequality and (4) for the equation in the middle row, 1 |uK x|K−τ C

=

1 C

≤

∞ X

! 21 2(K−τ )αn

e

2

|(x, fn )|

n=0

∞ X

! 21 e2Kβn |(x, fn )|2

= |x|K

n=0

≤

∞ X

! 21 2Kαn

e

2

|(x, fn )|

= |uK x|K .

n=0

By use of Lemma 9, applied to uK and its inverse, we obtain |x|t ≤ |uK x|t ≤ C|x|t+τ for all 0 ≤ t ≤ K − τ . α For every k ∈ N the set {uK : K ≥ k + τ } is an equicontinuous subset of L(Λα k+τ , Λk ). Since α α α α Λt −→ Λs is compact for t > s the set is relatively compact in L(Λk+τ +1 , Λk−1 ) for every k ∈ N. Therefore we may, by use of a diagonal procedure, find a subsequence uKn , so that uKn converges in −1 α L(Λα k+τ +1 , Λk−1 ) for every k ∈ N. Since the same applies to (uK )K we may choose the subsequence so α α that also (u−1 Kn )n converges in L(Λk+1 , Λk−1 ) for all k ∈ N, and we set for x ∈ Λ∞ (α): U x = lim uKn x, n→∞

V x = lim u−1 Kn x. n→∞

and certainly U, V ∈ L(Λ∞ (α)). Of course, we first take the limits in the local Banach spaces separately and then see that those results define elements U x ∈ Λ∞ (α), V x ∈ Λ∞ (α), respectively. It can easily be seen that U V = V U = id, hence U is an automorphism. We have |U x|0 = lim |Ukn x|0 = kxk n

345

D. Vogt

and we have for any t > 0 |x|t ≤ lim |Ukn x|t = |U x|t ≤ C|x|t+τ . n

This proves the result.



An endomorphism U of Λ∞ (α) for which there are constants C > 0 and τ ≥ 0 so that |U x|t ≤ C |x|t+τ ,

x ∈ Λ∞ (α)

for all t > 0 we call uniformly tame. A description of uniformly tame endomorphisms is contained in the following lemmas. For any endomorphism U ∈ L(Λ∞ (α)) there is a matrix (uk,j )k,j∈N0 so that   ∞ X Ux =  uk,j xj  , x ∈ Λ∞ (α). j=0

k∈N0

Of course, we have uk,j = hU ej , ek i if ek and ej denote the canonical basis vectors and h , i the `2 -scalar product. Lemma 11 Let α be strictly increasing and U ∈ L(Λ∞ (α)) be uniformly tame, then its matrix is upper triangular. P ROOF.

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 all t > 0 which implies the result.



If α is not necessarily strictly increasing we have to replace upper triangular by blockwise upper triangular, where the blocks are given by the sets of indices on which α is constant. Lemma 12 Let α be strictly increasing and S ∈ L(`2 ) have an upper triangular matrix, then S ∈ L(Λ∞ (α)) and S is uniformly tame with τ = 0. P ROOF. We may assume that kSkL(`2 ) = 1. Then in particular |sj,j | ≤ 1 for all j. Let (d0 , d1 , . . . ) with supj |dj | ≤ 4 be a sequence so that |sj,j + dj | = 4 for all j. We set Dx = (d0 x0 , d1 x1 , . . . ) and H = S + D. Let H0 x = (ho,o x0 , h1,1 x1 , . . . ) be the diagonal part of H and H+ = H − H0 then H+ = S+ where S0 and S+ are defined in an analogous way. Therefore kH+ k = kS+ k ≤ kSk + kS0 k ≤ 2, all norms taken in L(`2 ). Because of H = H0 (I + H0−1 H+ ) and kH0−1 H+ k ≤ 12 the operator H is invertible in L(`2 ) and −1 kH k ≤ 1. So we have |x|0 ≤ |Hx|0 ≤ 5|x|0 . We apply Lemma 10 to kxk = |Hx|0 and get a uniformly tame automorphism U of Λ∞ (α) with |x|t ≤ |U x|t ≤ 5|x|t ,

x ∈ Λ∞ (α)

for all t > 0, so that |U x|0 = |Hx|0 for all x. By Lemma 11 the matrix of U is upper triangular. Therefore V = U ◦ H −1 is unitary in `2 and has an upper triangular matrix. This implies that V is diagonal and |vj,j | = 1 for all j and therefore |Hx|t = |V −1 U x|t ≤ 5|x|t for all x ∈ Λ∞ (α) and t > 0. Finally we obtain |Sx|t ≤ |Hx|t + |Dx|t ≤ 9|x|t , x ∈ Λ∞ (α) for all t > 0. This proves the result.



Since we can modify any α to a strictly increasing sequence without changing Λ∞ (α) we obtain from Lemma 12 as an immediate consequence: 346

Power series spaces of infinite type

Lemma 13 Let S ∈ L(`2 ) have an upper triangular matrix, then S ∈ L(Λ∞ (α)). 

3.

Subspaces of power series spaces of infinite type

We characterize in this section the subspaces of a stable power series space Λ∞ (α). We assume that that the Fr´echet-Hilbert space E has property (DN) and is α-nuclear. By Corollary 1 the space E is isomorphic to a subspace of Σ∞ . We may assume that it is a subspace of Σ∞ . By | |t we denote the canonical norms on Σ∞ and also their restrictions to E. We assume chosen a sequence 0 = t0 < t1 < . . . with limk tk = ∞. We set k kk = | |tk on E and Uk = {x ∈ E : kxkk ≤ 1}. By definition the k k0 ≤ k k1 ≤ . . . are Hilbertian norms on E so that with suitable 0 < τk < 1 1−τk k k kk ≤ k kk−1 k kτk+1 .

(6)

By the choice of our seminorms the space E is countably normed, i.e. the canonical map ılk : Ek −→ El is injective for all k ≥ l. The fundamental lemma in this section is: Lemma 14 Assume that δn (Uk , U0 ) ≤ e−kαn for all n ≥ n0 . Then there is a sequence β with βn ≥ an for all n ≥ n0 and an injective map ϕk ∈ L(E, Λ∞ (β)), so that for all x ∈ E 1. |ϕk (x)|0 ≤ 2kxk0 , 2. |ϕk (x)|k ≥

1 2

kxkk .

Moreover ϕk extends to an isomorphism ϕˆk : E0 −→ `2 = Λβ0 . (s)

P ROOF. We set s = 2tk , E(s) the local Banach space of | |s and ıl : El −→ E(s) , ıl(s) : E(s) −→ El the respective linking maps. By assumption the map ı0k , hence also ı0(s) : E(s) −→ E0 is compact. Let sn be the singular numbers of ı0(s) . We set 1 βn = − log sn . 2k Then the Schmidt representation takes the form ı0(s) x =

∞ X

e−2kβn hx, en is fn .

(7)

n=0

Here (en )n is an orthonormal basis of E(s) and (fn )n an orthonormal basis of E0 . The choice of s and (2) applied to Σ∞ implies k k2k ≤ k k0 | |s . If δn (Uk , U0 ) ≤ e−kαn we obtain, by use of Lemmas 1 and 3, with U(s) = {x ∈ E : |x|s ≤ 1} e−2kβn = δn (U(s) , U0 ) ≤ δn2 (Uk , U0 ) ≤ e−2kαn . Therefore αn ≤ βn for all n ≥ n0 . We set ϕx = (hx, fn i0 )n∈N0 and obtain a unitary map ϕ : E0 −→ `2 = Λβ0 . From (7) we see that ϕ ◦ ı0(s) (x) = (e−2kβn hx, en is )n which means that ϕ ◦ ı0(s) is a unitary map E(s) −→ Λβ2k . We denote by u its inverse. Then u is unitary Λβ2k −→ E(s) and we have |u ξ|s =|ξ|2k kı0(s) u ξk0

=|ξ|0 347

D. Vogt

for all ξ ∈ Λβ2k . The latter because obviously ı0(s) ◦ u = ϕ−1 |Λβ .

(8)

2k

By Lemma 9, applied for maps between the Hilbert spaces of Λβt and Σt we 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 have kı0(s) u ξk0 = |uξ|0 and kık(s) u ξkk = |uξ|tk . We apply Lemma 8 (with ε = 21 ) to ϕ ◦ ı0(s) ∈ L(E(s) , Λβ2k ). We obtain maps ϕk ∈ L(E, Λ∞ (β)), ψ ∈ L(E0 , Λβk ) so that supkxk0 ≤1 |ψx|k ≤

1 2

and ϕ ◦ ı 0 = ϕk + ψ ◦ ı 0 .

We have to verify the desired properties for ϕk . For x ∈ E we have |ϕk x|0 ≤|ϕ(ı0 x)|0 + |ψ(ı0 x)|k ≤2kxk0 . This proves part 1. Moreover ϕk extends to ϕˆk ∈ L(E0 , Λβ0 ) and we have ϕˆk = ϕ − ψ = ϕ(I − ϕ−1 ψ). Since kϕ−1 ψkL(E0 ) ≤ 21 the map ϕˆk is invertible. The map ı0 is injective, hence also ϕk = ϕˆk ◦ ı0 . We may write ϕˆ−1 k by its Neumann series −1 ϕˆ−1 + ϕ−1 ψϕ−1 k =ϕ

∞ X

(ψϕ−1 )n

n=0

= ϕ−1 + ϕ−1 ◦ v where v=ψ

∞ X

(ϕ−1 ψ)n ϕ−1 .

n=0

Hence v ∈ L(Λβ0 , Λβk ) and kvkL(Λβ ,Λβ ) ≤ 1. 0 k For x ∈ E and ξ = ϕk x we obtain ı0 x = ϕ−1 (ξ) + ϕ−1 (vξ) = ϕ−1 (ξ + vξ).

(10)

Because of (9) the map ık(s) ◦ u extends to uk ∈ L(Λβk , Ek ). Due to (8), for η ∈ Λβk we have ϕ−1 η = ı0k uk η. With η = ξ + vξ we obtain from (10), using that ı0k is injective, ık x = uk (ξ + vξ). Therefore we have kxkk ≤ |ξ + vξ|k ≤ 2|ξ|k . This completes the proof.



< d < +∞ and βn ≥ αn for all n ≥ n0 , then there exists a strictly Lemma 15 If lim supn α2n+1 αn monotonous sequence (n(ν))ν∈N0 and b ≥ 0 so that βν ≤ αn(ν) ≤ b + dβν for all ν ∈ N0 . 348

Power series spaces of infinite type

P ROOF. We set m(ν) = sup{n : αn ≤ βν } and n(ν) = m(ν) + ν + 1. Then by assumption we have m(ν) ≥ ν for ν ≥ n0 and n(ν) is strictly increasing. Moreover βν < αn(ν) for all ν. We choose n1 ≥ n0 so that α2n+1 ≤ dαn for n ≥ n1 . For ν ≥ n1 we have αn(ν) ≤ α2m(ν)+1 ≤ dαm(ν) ≤ dβν . Setting b = sup{αn(ν) : ν < n1 } we obtain the result.



Lemma 16 Under the assumptions of Lemma 14 and the assumption that lim supn for every k there is an injective map ϕk ∈ L(E, Λ∞ (α)) so that for all x ∈ E

α2n αn

< +∞ we obtain:

1. |ϕk (x)|0 ≤ 2kxk0 2. |ϕk x|k ≥ 21 kxkk . P ROOF. We choose β and ϕk as in Lemma 14. We use Lemma 15 to find a sequence n(ν) with βν ≤ αn(ν) ≤ b + dβν and define for ξ = (ξ0 , ξ1 , . . .) a sequence ϕ ξ by ( ξν if n = n(ν) (ϕ ξ)n = 0 otherwise. Then ϕ gives an isomorphic imbedding Λ∞ (β) ,→ Λ∞ (α) and an isometric imbedding `2 ,→ `2 . If we replace ϕk by φk := ϕ ◦ ϕk , then φk ∈ L(E, Λ∞ (α)) is injective, we have |φk (x)|0 ≤ |ϕk (x)|0 ≤ 2kxk0 and with ξ = ϕk x ! 21 |φk x|k =|ϕ(ϕk x)|k =

X

|ξν |2 e2kαn(ν)

ν

! 12 ≥

X

|ξν |2 e2kβν

= |ϕk x|k ≥

ν

1 kxkk .  2

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 Let α be stable. A Fr´echet space E is isomorphic to a subspace of Λ∞ (α) if and only if E is an α-nuclear Fr´echet-Hilbert space with property (DN). P ROOF. If E is isomorphic to a subspace of Λ∞ (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 set n(i, k) = 2k + i2k+1 − 1 and get a bijective map N0 × N0 −→ N0 . We set αi,k := αn(i,k) and obtain an isomorphism Λ∞ (α) ∼ = Λ where Λ := {x = (xi,k )i,k∈N0 : kxk2t =

X

|xi,k |2 e2tαi,k < +∞ for all t ∈ R}.

i,k

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 every k ∈ N0 we choose ϕk ∈ L(E, Λ∞ (α)) according to Lemma 16 and set ϕk (x) = (ϕi,k (x))i∈N0 . 349

D. Vogt

We define for x ∈ E φ(x) := (2−k e−kαi,k ϕi,k (x))i,k∈N0 . We have to show that φ(x) ∈ Λ and φ ∈ L(E, Λ). From the stability of α we get bk ≥ 0, dk ∈ N so that αi ≤ αi,k ≤ bk + dk αi . We may assume that bk ≤ bk+1 and dk ≤ dk+1 for all k. Now, we fix m and choose M, C such that |ϕk x|mdm ≤ CkxkM for k = 0, . . . , m − 1. We get kφxk2m ≤

m−1 X

4−k

≤

! |ϕi,k (x)|2 e2(m−k)αi,k

+

i=0

k=0 m−1 X

∞ X

∞ X

4−k

k=m ∞ X

4−k e2mbm |ϕk (x)|2mdm +

k=0

∞ X

! |ϕi,k (x)|2

i=0

4−k |ϕk (x)|20

k=m

4 ≤ (e2mbm C 2 + 4)kxk2M . 3 On the other hand we have for m ∈ N −m

kφxk2m ≥ 2

∞ X

! 21 2 2mαi,m

|ϕi,m (x)| e

≥ 2−m |ϕm (x)|m ≥ 2−m−1 kxkm .

i=0

Therefore φ is an isomorphic imbedding E ,→ 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 space E has property (Ω) and is α-nuclear. By Corollary 1 the space E is isomorphic to a quotient of Σ∞ . Let q : Σ∞ −→ E be the quotient map. By | |t we denote the quotient seminorms under q of the canonical norms | |t on Σ∞ . We assume chosen a sequence 0 = t0 < t1 < . . . with limk tk = ∞. We set k kk = | |tk and Uk = {x : kxkk ≤ 1}. By definition the k k0 ≤ k k1 ≤ . . . are Hilbertian seminorms on E so that with suitable 0 < ϑk < 1 and Ck > 0 1−ϑk ϑk k k∗k ≤ Ck k k∗k−1 k k∗k+1 . (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 ≥ αn for all n ≥ n0 and a map ϕk ∈ L(Λ∞ (β), E) so that for all ξ ∈ Λ∞ (β) 1. kϕk ξkk ≤ 2|ξ|0 2. kϕ ξkk+1 ≥ 12 |ξ|p . Moreover ϕk extends to an isomorphism ϕˆk : `2 = Λβ0 −→ Ek . 350

Power series spaces of infinite type

P ROOF.

By assumption the map ıkk+1 is compact. We set βn = −

1 log δn (Uk+1 , Uk ). 2p

Then the Schmidt representation takes the form ıkk+1 x =

∞ X

e−2pβn hx, en ik+1 fn

n=0

where (en )n∈N0 is an orthonormal sequence in Ek+1 , (fn )n∈N0 an orthonormal basis of Ek . As in the proof of Lemma 14, we get αn ≤ βn for all n ≥ n0 . We set ϕξ =

∞ X

ξn e2pβn en

n=0

and obtain an isometry ϕ from Λβ2p into Ek+1 . Because of ıkk+1 ϕξ =

∞ X

ξn fn

n=0

the map ıkk+1 ◦ ϕ extends to a unitary map χ : `2 = Λβ0 −→ Ek . (s)

We put s = 21 (tk+1 − tk ). We denote by E(s) the local Hilbert space of | |s and by ı(s) , ık+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 on E. We obtain ϕk ∈ L(Λ∞ (β), E), ψ ∈ L(Λ∞ (β), E(s) ) so that sup|ξ|0 ≤1 |ψξ|s ≤ 12 and (s)

ık+1 ◦ ϕ = ı(s) ◦ ϕk + ψ. We have to verify the desired properties for ϕk . We have ık ◦ ϕk = ıkk+1 ◦ ϕ − ık(s) ◦ ψ. Therefore kϕk ξkk ≤ 2|ξ|0 for ξ ∈ E. Moreover ık ◦ ϕk extends to a map ϕˆk ∈ L(`2 , Ek ) which can be written as ˆ ϕˆk = χ ◦ (id −χ−1 ◦ ık(s) ◦ ψ). ˆ L(` ) ≤ Here ψˆ denotes, as always, the continuous extension. Since χ is invertible and kχ−1 ◦ ık(s) ◦ ψk 2 the map ϕˆk is invertible. We have for x ∈ Ek χ−1 (x) = (hx, fn ik )n∈N0 . Therefore for x ∈ Ek+1

1 2

χ−1 (ıkk+1 x) = (e−2pβn hx, en ik+1 )n∈N0 .

For x ∈ E we define h(x) := χ−1 (ık x) and obtain |h(x)|0 ≤kxkk |h(x)|2p ≤kxkk+1 . Therefore by Lemma 9, applied to Σ∞ and Λ∞ (β), we get |h(x)|p ≤|x|s . 351

D. Vogt

This implies that χ−1 ◦ ık(s) ◦ ψ extends to a map T ∈ L(Λβ0 , Λβp ) and we have kT kL(Λβ ,Λβp ) ≤ −1

L(Λβp ),

0

1 2.

Therefore id −T is invertible in with k(id −T ) kL(Λβp ) ≤ 2. For ξ ∈ Λ∞ (β) we set x = (id −T )ξ and have ϕk (ξ) = χ(x) hence x = h(ϕk (ξ)) which implies |ξ|p ≤ 2|x|p ≤ 2kϕk (ξ)kk+1 . This completes the proof.



We assume now that α is stable and may choose d ∈ N, so that lim sup n

α2n+1 < d. αn

Lemma 18 Under the assumptions of Lemma 17 and the assumption that δn (Uk+1 , Uk ) ≤ e−2dpαn for n ≥ n0 and lim supn α2n+1 αn < d we obtain: there is a map ϕk ∈ L(Λ∞ (α), E) and ck > 0, so that 1. kϕk ξkk ≤ 2|ξ|0 for all ξ ∈ Λ∞ (α) 2. |ϕ0k y|∗p ≥ ck kyk∗k+1 for all y ∈ E 0 . Moreover ϕk extends to a surjection ϕˆk : Λα 0 −→ Ek . P ROOF. From Lemma 17, applied to (dαn )n , we get for given k a sequence β with βn ≥ dαn for n ≥ n0 and a map ϕk ∈ L(Λ∞ (β), E) with the properties 1. and 2. in Lemma 17. From Lemma 15 we get a increasing sequence (n(ν))ν∈N0 and b > 0 so that 1 βν ≤ αn(ν) ≤ b + βν . d We set, as in the proof of Lemma 16 ( ξν (ϕξ)n = 0

if n = n(ν) otherwise.

Then ϕ gives an isomorphic imbedding Λ∞ (β) ,→ Λ∞ (α) and an isometric imbedding `2 ,→ `2 . We have |ϕx|t ≤ etb |x|t for all t ∈ R. We define L ∈ L(Λ∞ (α), Λ∞ (β)) by (Lξ)ν∈N0 = (ξn(ν) )ν∈N0 . Then L ◦ ϕ = id and |Lξ|t ≤ |x|dt for all t ≥ 0. We put φk = ϕk ◦ L. Then φk ∈ L(Λ∞ (α), E). Moreover 1. is satisfied since |Lξ|0 ≤ |ξ|0 and, because L is surjective, also φˆk is surjective. As for 2. we notice that for y ∈ E 0 1 kyk∗k+1 ≤|ϕ0k y|∗p = |(ϕ0 ◦ L0 ◦ ϕ0k )y|∗p 2 ≤epb |(L0 ◦ ϕ0k )y|∗p = epb |φ0k y|∗p .  We are now ready to prove the main theorem of this section. For the nuclear case see [30, Satz 3.4]. Theorem 4 Let α be stable. A Fr´echet space E is isomorphic to a quotient space of Λ∞ (α) if and only if E is an α-nuclear Fr´echet-Hilbert space with property (Ω). 352

Power series spaces of infinite type

P ROOF. If E is isomorphic to a quotient space of Λ∞ (α) then it is Fr´echet-Hilbert and, by Proposition 4, it is α-nuclear and has property (Ω). We have to show the converse implication. As in the proof of Theorem 3 we set n(i, k) = 2k +i2k+1 −1 and αi,k := αn(i,k) and use the isomorphic representation Λ∞ (α) ∼ = Λ where X Λ := {x = (xi,k )i,k∈N0 : kxk2t = |xi,k |2 e2tαi,k < +∞ fore all t ∈ R}. i,k

For every k ∈ N0 we choose dk ∈ N, bk ≥ 0 so that αi,k ≤ bk + dk αi for all i and set pk = kdk . We choose by induction a sequence t0 = 0 < t1 < t2 < . . . so that the seminorms k kk = | |tk satisfy the assumptions of Lemma 18 with p = pk . For each k ∈ N0 let ψk be the map from Lemma 18. For x = (xi,k )i,k∈N0 and k ∈ N0 set xk = (xi,k )i∈N0 . Since i ≤ n(i, k), hence αi ≤ αn(i,k) = αi,k , we have xk ∈ Λ∞ (α) and ∞ ∞ X ∞ X X |xk |2t ≤ |xi,k |2 e2tαi,k = kxk2t k=0 i=0

k=0

for all t ≥ 0. We put Ψ(x) :=

∞ X

k

2− 2 ψk (xk )

k=1

for x = (xk )k = (xi,k )i,k ∈ Λ. We show that the series converges and defines Ψ ∈ L(Λ, E). This follows from !2 m−1 ∞ X X k 2 −k − kΨxkm ≤ 2 2 kψk xk km + 2 2 kψk xk kk k=1

≤C 2

m−1 X

k=m

|xk |2M + 4

k=1

≤(C 2 + 4)

∞ X

|xk |20

k=m ∞ X

|xk |2M

k=0

≤(C 2 + 4)kxk2M with suitable C and M depending from m. Since ψk ∈ L(Λ∞ (α), E) and ψˆk is surjective from `2 onto Ek , we see that R(Ψ) is dense in E. Let m be given. We obtain for y ∈ E 0 : X m 0 (y))(ξ)| : kΨ0 yk∗m ≥2− 2 sup{|(ψm |ξi |2 e2mαi,m ≤ 1} i m

0 ≥2− 2 sup{|(ψm (y))(ξ)| : |ξ|mdm ≤ e−mbm } m

0 (y)|∗mdm ≥2− 2 · e−mbm |ψm m

≥2− 2 e−mbm cmdm kyk∗m+1 . The surjectivity criterion [12, 26.1] then shows that Ψ is surjective.

5.



Complemented subspaces of power series spaces of infinite type

Our next task is to characterize the complemented subspaces of Λ∞ (α) by invariants and study their structure. 353

D. Vogt

First we prove a partial replacement for the Ramanujan and Terzio˘glu imbedding theorem from [17]. Lemma 19 If Q is a quotient of Λ∞ (α), equipped with the quotient seminorms k kt of the norms | |t , t ∈ R, then for any k there is Sk ∈ L(Q, Λ∞ (α)), so that supkxkk ≤1 |Sk x|0 < +∞ and Sk induces an isometry Sˆk : Qk ,→ Λ00 = `2 . P ROOF. We may assume k = 0 and dim Q0 = ∞. Let ϕ : Λ∞ (α) −→ Q be the quotient map. For K > 0 the map ı0K : QK −→ Q0 is compact. Let sn be its singular numbers. We set βn = −

1 log sn . K

Then its Schmidt representation takes the form X ı0K x = e−Kβn hx, en iK fn , n

where (en )n∈N0 is an orthonormal system in QK , (fn )n∈N0 an orthonormal basis of Q0 . By Remark 2 and Lemma 2 we get e−Kβn ≤ e−Kαn i.e. βn ≥ αn for all n. We set for x ∈ Λ∞ (α) TK x := (hx, fn i)n . Then we have |TK x|β0 =kxk0 ,

|TK x|βK =kxkK ,

|TK x|α 0 =kxk0 ,

|TK x|α K ≤kxkK .

hence

and therefore by Lemma 9, applied to TK ◦ q, |Tk x|α k ≤ kxkk for all 0 ≤ k ≤ K. Here | |βt and | |α t denote the norms in Λ∞ (β) and Λ∞ (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 that TKn x −→ T x for all x ∈ Q. For every x ∈ Q we have |T x|0 = lim |TKn x|0 = kxk0 . n

Setting S0 := T we obtain the result.



As an immediate consequence we obtain: Lemma 20 If Q is a quotient of Λ∞ (α) then there is an imbedding S : Q ,→ Λ∞ (α)N . P ROOF.

We set Sx = (Sk x)k∈N , Sk as in Lemma 19.



We can now give the characterization of the complemented subspaces of Λ∞ (α). For the nuclear case see [30, Satz 3.5]. Theorem 5 Let α be stable. A Fr´echet space E is isomorphic to a complemented subspace of Λ∞ (α) if and only if E is an α-nuclear Fr´echet-Hilbert space with properties (DN) and (Ω). 354

Power series spaces of infinite type

P ROOF. If E is isomorphic to a complemented subspace of Λ∞ (α) then it is Fr´echet-Hilbert and by Proposition 4, it is α-nuclear and has properties (DN) and (Ω). We have to prove the converse implication. Since E is an α-nuclear Fr´echet-Hilbert space with property (DN) we know from Theorem 3 that E is isomorphic to a subspace of Λ∞ (α), this means that there is an isomorphic imbedding j : E ,→ Λ∞ (α). We set Q := Λ∞ (α)/jE. By Lemma 20 there is an imbedding S : Q ,→ Λ∞ (α)N . We consider the exact sequence ϕ

0 −−−−→ Λ∞ (α) −−−−→ Λ∞ (α) −−−−→ Λ∞ (α)N −−−−→ 0 ˜ = ϕ−1 (SQ). We can set up the following commutative diagram with exact from Proposition 1 and set Q rows and columns: 0 x   j

0 x   q

0 −−−−→ E −−−−→ Λ∞ (α) −−−−→ x x ϕ0 =   0 −−−−→ E −−−−→

H x  

q0

−−−−→

Q x ϕ 

−−−−→ 0

˜ Q x  

−−−−→ 0

=

Λ∞ (α) −−−−→ Λ∞ (α) x x     0

0

˜ : qx = ϕy} is a Fr´echet-Hilbert space, ϕ0 (x, y) = x, q0 (x, y) = y. Here H = {(x, y) ∈ Λ∞ (α) × Q ˜ as a subspace of Λ∞ (α) has property (DN) the second Since by assumption E has property (Ω) and Q row splits by Theorem 2. Since Λ∞ (α) has properties (DN) and (Ω) the first column splits by the same theorem. Therefore we have ˜∼ E⊕Q =H∼ = Λ∞ (α) ⊕ Λ∞ (α) ∼ = Λ∞ (α). The last isomorphism exists since α is stable.

6.



About the structure of complemented subspaces of power series spaces of infinite type

After having characterized the complemented subspaces of power series spaces by invariants we will now closer study their structure. In [14, Theorem14] (see also [13]) Mityagin proved that every complemented subspace of a finite type power series space has a basis and is again isomorphic to a finite type power series space. He posed the problem whether the same is true for infinite type power series spaces, see [14, Problem 15]. In [15] he showed that that every complemented subspace of an infinite type power series space with an unconditional basis is isomorphic to an infinite type power series space. So the problem remains whether every complemented subspace of an infinite type power series space has an unconditional basis. This problem is, even in the nuclear case, still unsolved. A solution for the nuclear case has been announced without proof in Kondakov [10]. A proof proposed by the same author in [8] is apparently mistaken. Nevertheless there are many partial solutions. We will describe some of them. A principal tool for that is the following result, in the nuclear case it was first proved in [23]. 355

D. Vogt

Lemma 21 If α is stable and E is a complemented subspace of Λ∞ (α), then E ⊕ Λ∞ (α) ∼ = Λ∞ (α). P ROOF.

Let F be a complement of E in Λ∞ (α), i.e. E ⊕ F ∼ = Λ∞ (α), then Λ∞ (α)N ∼ = EN ⊕ F N ∼ = E ⊕ EN ⊕ F N ∼ = E ⊕ Λ∞ (α)N .

We use again the exact sequence from Proposition 1. We add the exact sequence id

0 −−−−→ 0 −−−−→ E −−−−→ E −−−−→ 0 and obtain by use of the previous isomorphism 0 −−−−→ Λ∞ (α) −−−−→ Λ∞ (α) ⊕ E −−−−→ Λ∞ (α)N −−−−→ 0. Like in the proof of Theorem 5 we obtain the following diagram 0 x  

0 x  

0 −−−−→ Λ∞ (α) −−−−→ Λ∞ (α) −−−−→ x   0 −−−−→ Λ∞ (α) −−−−→

H x  

Λ∞ (α)N x  

−−−−→ 0

−−−−→ Λ∞ (α) ⊕ E −−−−→ 0 x  

Λ∞ (α) x  

Λ∞ (α) x  

0

0

Here H ⊂ Λ∞ (α) ⊕ Λ∞ (α) ⊕ E is a Fr´echet-Hilbert space. By use of Theorem 2 the first column and the second row split and we get Λ∞ (α) ⊕ E ∼ = Λ∞ (α) ⊕ Λ∞ (α) ⊕ E ∼ =H∼ = Λ∞ (α) ⊕ Λ∞ (α) ∼ = Λ∞ (α). Here we used several times the stability of α.



In the nuclear case the following theorem was first proved in [23]. Theorem 6 Let α be stable. If E is isomorphic to a complemented subspace of Λ∞ (α) and Λ∞ (α) isomorphic to a complemented subspace of E, then E ∼ = Λ∞ (α). P ROOF. Let E ⊕ F ∼ = Λ∞ (α) and G ⊕ Λ∞ (α) ∼ = E, then G ⊕ Λ∞ (α) ⊕ F ∼ = Λ∞ (α). Therefore G is isomorphic to a complemented subspace of Λ∞ (α). By Lemma 21 we have E∼ = G ⊕ Λ∞ (α) ∼ = Λ∞ (α).  We will now present sufficient conditions so that a Fr´echet-Hilbert space with properties (DN) and (Ω) has a basis. To formulate them we need one more concept which is due Aytuna-Krone-Terzio˘glu [2]. Let E be a Fr´echet-Hilbert-Schwartz space with properties (DN) and (Ω). Let k kp be a dominating norm and choose, for given p, a q according to (Ω). We set Uk = {x : kxkk ≤ 1} for all k. Definition 4 The sequence αn := − log δn (Uq , Up ) is called an associated exponent sequence of E and Λ∞ (α) the associated power series space. 356

Power series spaces of infinite type

From [2] we know the following: Lemma 22 Λ∞ (α) depends only on E, not on the choice of Up , Uq .  By use of Lemma 3 this implies that, in particular, E is α-nuclear. As a consequence of Lemma 22 we obtain: Corollary 4 If E is a Fr´echet-Hilbert-Schwartz space with properties (DN) and (Ω) and Λ∞ (α) its associated power series space and if E ∼ = F then F has the same properties, in particular Λ∞ (α) is its associated power series space.  This yields Proposition 5 If λ(A) := {x = (xn )n∈N0 : kxk2k :=

∞ X

|xj |2 aj,k < +∞ for all k}

j=0

has properties (DN) and (Ω), and Λ∞ (α) is its associated power series space then λ(A) ∼ = Λ∞ (α). P ROOF. By [29, Satz 2.7] there is a power series space Λ∞ (β) so that λ(A) ∼ = Λ∞ (β). Since then clearly Λ∞ (β) is the associated power series space of λ(A), we have by Corollary 4 Λ∞ (β) = Λ∞ (α).  This means that, if a Fr´echet-Hilbert-Schwartz space with properties (DN) and (Ω) is isomorphic to a K¨othe space λ(A) defined as above, then it is isomorphic to its associated power series space. Since, by Theorem 5 with suitable α, it is always isomorphic to a complemented subspace of some Λ∞ (α) the result of Mityagin [15] shows that this is always the case if it has an unconditional basis. We will use a simplified version of the proofs of Lemma 14 and Lemma 17 to show the following. Lemma 23 Let E be a Fr´echet-Hilbert-Schwartz space with properties (DN) and (Ω). Let k k0 be a Hilbertian dominating norm, k k1 chosen for k k0 according to (Ω) and αn = − log δn (U1 , U0 ) where Uj = {x ∈ E : kxkj ≤ 1}. Then there exist maps ψ ∈ L(Λ∞ (α), E), ϕ ∈ L(E, Λ∞ (α)) so that ψ extends to an isomorphism ψ0 : `2 −→ E0 , ϕ extends to an isomorphism ϕ0 : E0 −→ `2 and we have sup |ξ − ϕ0 ◦ ψ0 (ξ)|0 < |ξ|0 ≤1

P ROOF.

1 . 2

By assumption ı01 is compact and its Schmidt representation takes the form ı01 x =

∞ X

e−αn hx, en i1 fn

n=0

where (en )n is an orthonormal sequence in E1 and (fn )n an orthonormal basis of E0 . We set ϕ(x) ˜ = (hx, fn i0 )n∈N0 and obtain a unitary map ϕ˜ : E0 −→ `0 = Λα 0 for which we have ϕ˜ ◦ ı01 (x) = (e−αn hx, en i1 )n∈N0 ⊥ which means that ϕ◦ı ˜ 01 defines an isometry ϕ˜1 from F := span{e0 , e1 , . . .} onto Λα = ker ϕ◦ı ˜ 01 = 1 with F α ˜ ker ı01 . We set ψ˜ = ϕ˜−1 . Then ψ˜ is a unitary map `2 = Λα −→ E so that there is a map ψ : Λ −→ E1 0 1 0 1

357

D. Vogt

˜ We apply Lemma 8 to ϕ˜1 and ψ˜1 and obtain maps ϕ ∈ L(E, Λ∞ (α)), ψ ∈ L(Λ∞ (α), E) with ı01 ◦ ψ˜1 = ψ. so that with 0 < ε < 1 to be determined later ϕ˜ ◦ ı0 =ϕ + χ ◦ ı0 ψ˜ =ı0 ◦ ψ + η

on E on Λ∞ (α)

α where χ ∈ L(E0 , Λα 0 ) with supkxk0 ≤1 |χ(x)|0 < ε and η ∈ L(Λ0 , E0 ) with sup|ξ|0 ≤1 kη(ξ)k0 < ε. Therefore ϕ extends to ϕ0 = ϕ˜ − χ ∈ L(E0 , `2 ) and ψ to ψ0 = ψ˜ − η ∈ L(`2 , E0 ). Since kχk < 1 and kηk < 1 ϕ0 and ψ0 are invertible. Moreover ϕ0 ◦ ψ0 = id −χ ◦ ψ˜ − ϕ˜ ◦ η + χ ◦ η

and therefore sup |ξ − ϕ0 ◦ ψ0 (ξ)|0 < 2ε + ε2 < |ξ|0 ≤1

for small ε > 0.

1 2



Corollary 5 Under the assumptions of Lemma 23 there exist maps ψ ∈ L(Λ∞ (α), E), ϕ ∈ L(E, Λ∞ (α)) so that ψ extends to a unitary map ψ0 : `2 −→ E0 and ϕ extends to a unitary map ϕ0 : E0 −→ `2 . P ROOF. We choose ψ and ϕ according to Lemma 23. Then we apply Lemma 10 to the norm kxk = kψxk0 and obtain an automorphism U of Λ∞ (α) so that |U x|0 = kψxk0 . We do the same with the norm kxk = kϕ−1 xk0 and obtain an automorphism V of Λ∞ (α) so that |V x|0 = kϕ−1 xk0 . Finally we replace ψ by ψ ◦ U −1 and ϕ by V ◦ ϕ.  An important step now is contained in the following lemma. The method for the construction of S is due to Aytuna, Krone and Terzio˘glu [1], the difference here is again, that we don’t need nuclearity. Lemma 24 Let α be stable, T ∈ L(Λ∞ (α)) so that T induces a unitary map in L(`2 ). Then there is S ∈ L(Λ∞ (α)), so that P = T ◦ S is a projection in Λ∞ (α), orthogonal in `2 , and R(P ) ∼ = Λ∞ (α). P ROOF. Let ej = (0, . . . , 0, 1, 0, . . .) ∈ Λ∞ (α) and fj = T ej . We choose inductively vectors gn ∈ Λ∞ (α) with following properties: (1) gn ∈ span{f0 , . . . , f2n } (2) gn ⊥g0 , . . . , gn−1 in `2 (3) gn ⊥e0 , . . . , en−1 in `2 (4) |gn |0 = 1. This is possible since dim span{f0 , . . . , f2n } = 2n + 1. Due to (1) we have gn :=

2n X

µk,n fk = T (

k=0

2n X

µk,n ek ).

k=0

We set hn =

2n X

µk,n ek

k=0

and obtain an orthonormal system (hn )n∈N0 . We set µk,n = 0 for k > 2n. 358

Power series spaces of infinite type

We define

∞ X

Sx :=

hx, gn ihn .

n=0

This means S = T −1 ◦ P where P is the orthogonal projection onto span{g0 , g1 , . . .}. We have to show that S defines a map in L(Λ∞ (α)). We do that in two steps. First we define a map ϕ ∈ L(`2 ) by ϕ(x) =

∞ X

hx, gn ien .

n=0

For the matrix elements ϕk,j = hϕej , ek i = hej , gk i we have ϕk,j = 0 for k > j. Therefore, by Lemma 13, ϕ ∈ L(Λ∞ (α)). Next we define a map ψ ∈ L(`2 ) by ψ(x) =

∞ X

hx, en ihn .

n=0

For the matrix elements ψk,j = hψej , ek i = hhj , ek i we obtain that ψk,j = 0 for k > 2j. We define ψ˜ ∈ L(`2 ) by ψ˜ = ψ ◦ A and Ax = (x2n )n∈N0 for x = (xn )n∈N0 . Then we have  hψeν , ek i : j = 2ν ˜ hψej , ek i = 0 : j = 2ν + 1. ˜ j , ek i = 0 for k > j. By Lemma 13 we obtain that ψ˜ ∈ L(Λ∞ (α)). This means that ψ˜k,j = hψe Now we set  xν : j = 2ν (Bx)j = 0 : j = 2ν + 1. ˜ for x = (xn )n∈N0 . Due to the stability of α we have B ∈ L(Λ∞ (α)) and therefore ψ = ψ◦B ∈ L(Λ∞ (α)). Since obviously S = ψ ◦ ϕ we have shown that S ∈ L(Λ∞ (α)). It remains to show that R(P ) ∼ = Λ∞ (α). The map T ◦ψ ∈ L(Λ∞ (α), R(P )) is injective and, because of (T ◦ψ)◦ϕ = T ◦S = P , also surjective. Therefore it is an isomorphism.  From Lemmas 23, 24 and Theorems 5, 6 we derive the following theorem which was shown in the nuclear case by Aytuna, Krone and Terzio˘glu in [1]. Theorem 7 If E is a Fr´echet-Hilbert-Schwartz space with properties (DN) and (Ω) and its associated power series space Λ∞ (α) is stable, then E ∼ = Λ∞ (α). P ROOF. From Corollary 5 we get ϕ ∈ L(E, Λ∞ (α)), ψ ∈ L(Λ∞ (α), E) so that T := ϕ ◦ ψ extends to a unitary map in L(`2 ). Then by Lemma 24 we get S ∈ L(Λ∞ (α)), so that P = T ◦ S is a projection in Λ∞ (α) with R(P ) ∼ = Λ∞ (α). We set π := ψ ◦ S ◦ P ◦ ϕ ∈ L(E) and obtain a projection. P ◦ ϕ ∈ L(R(π), R(P )) is an isomorphism, since ψ ◦ S|R(P ) is its inverse. As the assumptions imply that E is α-nuclear (see the remark after Lemma 22), Theorems 5 and 6 yield the result.  Stability of the associated power series space is one condition which implies the existence of a basis in E, and by far the most important for analysis since in applications in analysis the associated power series usually can be calculated and is stable. Another condition is the following, see [6], [7]. Definition 5 Λ∞ (α) is called tame if, up to equivalence, α has the following form: there are strictly increasing sequences n(k) in N0 with n(0) = 0 and βk > 0 so that 359

D. Vogt

(1) αn = βk for n(k) ≤ n < n(k + 1) (2) limk

βk+1 βk

= +∞.

These spaces have the following properties (see [6, Proposition 1], [7, Theorem 1.3]). Theorem 8 The following are equivalent (1) Λ∞ (α) is tame. (2) There exists d so that for every A ∈ L(Λ∞ (α)) there is b with |Ax|k ≤ Ck |x|dk+b for all k with suitable Ck . (3) There are countably many functions σm (·), m ∈ N so that for every A ∈ L(Λ∞ (α)) there is m with |Ax|k ≤ Ck |x|σm (k) for all k with suitable Ck . α

(4) The set of finite limit points of { αµν : µ, ν ∈ N0 } is bounded.  If α has, without equivalence, the form given in the definition, then α is called blockwise unstable. Then we have in a natural way Y X `2 (m(k)) : |x|2t = |xk |2 e2tβk (12) Λ∞ (α) ∼ = {x = (x0 , x1 , . . .) ∈ k

k

< + ∞ for all t ∈ R} where m(k) = n(k + 1) − n(k) and `2 (m(k)) is the m(k)-dimensional Hilbert space. Every A ∈ L(Λ∞ (α)) corresponds to a matrix (Aj,ν )j,ν∈N0 where Aj,ν ∈ L(`2 (m(ν)), `2 (m(j))). We put A0j,ν := δj,ν Aj,ν , A1j,ν = Aj,ν − A0j,ν . The following result, which is a generalization of a result in [3], is contained in [7, Lemma 2.1]. Lemma 25 If α is blockwise unstable, A ∈ L(Λ∞ (α)) and |Ax|0 ≤ C|x|0 , then for each ε > 0 the set A1 Uε is relatively compact in Λ∞ (α), where Uε = {x ∈ Λ∞ (α) : |x|ε ≤ 1}. P ROOF. We need to prove only that A1 Uε is bounded in L(Λ∞ (α)). Fix t > 0. (1) For ν > j ≥ j0 we obtain from the continuity of A in `2   21 X ∞ X ∞ Aj,ν xν ≤ C  |xν |2  . ν=j+1 ν=j+1 Therefore

  12 X ∞ X ∞ tβ Aj,ν xν e j ≤ Ce−βj  |xν |2 e2εβν  ν=j+1 ν=j+1

if j0 is so large that (t + 1)βj ≤ εβj+1 . 360

Power series spaces of infinite type

(2) For ν < j, j ≥ j0 we find Ct+2 , σ(t + 2) so that |Ax|t+2 ≤ Ct+2 |x|σ(t+2) and therefore kAj,ν ke(t+2)βj ≤ Ct+2 eσ(t+2)βν . Here kAj,ν k denotes the norm in L(`2 (m(ν)), `2 (m(j))). From this we get kAj,ν ketβj ≤Ct+2 eσ(t+2)βν −2βj ≤Ct+2 e−βj if j0 is so large that σ(t + 2)βj−1 ≤ βj for j ≥ j0 . (3) From all this we get

|A1 x|t ≤etβj0

 12  2  2  12 ∞ ∞ j ∞ 0 X X   X X  Aj,ν xν  +  Aj,ν xν e2tβj  j=1 ν=0

j=j0 ν=j+1

 12 j−1 2 ∞ X X Aj,ν xν e2tβj  + 

j=j0 ν=0

 12  21   X X ≤etβj0 C|x|0 + C  e−2βj  |x|ε + Ct+2  j 2 e−2βj  |x|0 . j

j

The last estimate holds since j−1 ! 12 j−1 X X tβj Aj,ν xν e ≤ |x|0 kAj,ν k2 e2tβj ν=0

ν=0

1

≤Ct+2 j 2 e−βj |x|0 .  Lemma 26 Let Λ∞ (α) be tame, A ∈ L(Λ∞ (α)) and sup|x|0 ≤1 |x − Ax|0 < 1. Then A is invertible. P ROOF. We may assume that α is blockwise unstable and that, in the notation of Definition 5, all βj ∈ N. We calculate x − A0 x as follows: we put B(t)x = (eitβj x)j for x = (xj )j ∈ Λ∞ (α) in the representation (12). Then B(t) ∈ L(Λ∞ (α)) and B(t) is unitary on `2 . We obtain Z 2π 1 0 x−A x= B(t)(I − A)B(−t)x dt. 2π 0 From this we derive easily sup |x − A0 x|0 < 1. |x|0 ≤1 −1

0 Therefore A0 is invertible in `2 = Λα being a blockwise diagonal map defines a map in L(Λ∞ (α)). 0. A Hence we obtain −1 −1 A0 ◦ A = I + A0 ◦ A1 −1

−1

where A0 ◦ A1 is compact in L(Λ∞ (α)). This implies that A0 ◦ A is a Fredholm map with index = 0 in L(Λ∞ (α)). 0 −1 ◦ A = {0}. This proves The assumption implies that A is invertible in `2 = Λα 0 . Therefore ker A the result.  We obtain the following theorem, for the nuclear case see Wagner [32, Theorem 5] and Kondakov [9]. 361

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Theorem 9 If E is a Fr´echet-Hilbert-Schwartz space with properties (DN) and (Ω) and its associated power series space Λ∞ (α) is tame, then E ∼ = Λ∞ (α). P ROOF. We apply Lemma 26 to A = ϕ ◦ ψ where ϕ ∈ L(E, Λ∞ (α)) and ψ ∈ L(Λ∞ (α), E) are the maps of Lemma 23. Notice that α in Lemma 23 is the associated exponent sequence. Lemma 26 yields that A is invertible. We set χ := A−1 ◦ ϕ ∈ L(E, Λ∞ (α)). Then χ ◦ ψ = id. From this we conclude that P := ψ ◦ χ is a projection in E. If P (x) = 0 then |x|0 = 0 hence x = 0. So ker P = 0 and P = id, i.e. ψ ◦ χ = id. So χ is an isomorphism.  The preceding theorem is a generalization of the following theorem shown in [6, Theorem], [7, Theorem 2.4]. Theorem 10 If Λ∞ (α) is tame then every complemented subspace of Λ∞ (α) has a basis. P ROOF. We have to show that Theorem 9 implies Theorem 10. Let E be complemented in Λ∞ (α) and F a complement. Let Λ∞ (β) and Λ∞ (γ) be the associated power series spaces of E and F , respectively. Then clearly Λ∞ (α) = Λ∞ (β) ⊕ Λ∞ (γ). where ⊕ means that to get Λ∞ (α) we have to take an increasing common rearrangement of α and β. Therefore there is a subsequence α ˜ = (αjν )ν∈N0 so that Λ∞ (˜ α) = Λ∞ (β). From there it is easily seen that Λ∞ (β) is tame, hence E ∼ = Λ∞ (β) by Theorem 9. 

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D. Vogt FB Mathematik-Naturwissenschaften Bergische Universit¨at Wuppertal Gaussstrasse 20 D-42097 Wuppertal, Germany [email protected]

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