Strictly singular operators and isomorphisms of Cartesian products of power series spaces

V. P. Zahariuta, in 1973, used the theory of Fredholm operators to develop a method to classify Cartesian products of locally convex spaces. In this work we modify his method to study the isomorphic classification of Cartesian products of the kind E0…a† E1 …b† where 1 % p;q < 1 , p ˆj q, a ˆ …an† nˆ1 and b ˆ …bn† nˆ1 are sequences of positive numbers and E0…a†, E1 …b† are respectively `p-finite and `q-infinite type power series spaces. Introduction. Let …aik†i;k2N be a matrix of real numbers, such that 0 % aik % ai;k‡1 for all i; k and p ^ 1: We denote by K…aik† the `p-Köthe space defined by the matrix …aik†; i.e. the space of all sequences of scalars x ˆ …xi† such that jxjk :ˆ P i …jxijaik† 1=p < 1 8 k 2 N: With the topology generated by the system of seminorms fj:jk; k 2 Ng it is a FreÂchet space. If a ˆ …ai† is a sequence of positive numbers the Köthe spaces E0…a† ˆ K exp ÿ 1 k ai ; E1 …a† ˆ K…exp…kai†† are called, respectively, `p-finite and `p-infinite type power series spaces. They are Schwartz spaces if and only if ai ! 1 : Power series spaces play an important role in Functional Analysis because they provide sequence space representations for large classes of spaces of (analytic or C1 ) functions (see for more details [8, 9, 14]). Their isomorphic classification and structure properties were studied by Kolmogorov, Pelczynski, Mityagin and many other mathematicians and the question of isomorphic classification was solved completely in the Schwartzian case (see [8] for details) by the help of classical linear topological invariants, namely approximative and diametral dimensions. For arbitrary (non-Schwartzian) spaces, Mityagin [10] obtained a complete isomorphic classification of `2-power series spaces (see also [12]). Moreover, he initiated in this paper a Arch. Math. 70 (1998) 57±65 0003-889X/98/010057-09 $ 3.30/0 Birkhäuser Verlag, Basel, 1998 Archiv der Mathematik Mathematics Subject Classification (1991): 46A04, 47A99. *) Research supported by TÜBI . TAK-NATO Fellowship Program and partially by NRF of Bulgaria, grant MM-409/94. method to construct new (generalized) linear topological invariants that are more powerful than approximative and diametral dimensions. Zahariuta [18] developed this method for KoÈ the spaces and obtained new results about isomorphic classification for some classes of KoÈ the spaces that include Cartesian and tensor products of power series spaces (for further developments see the survey [22]). Another approach to the isomorphic classification of Cartesian products was Zahariuta s use of the theory of Fredholm operators [19, 20]. We modify Zahariuta s method (following [13], see also [21]) in order to extend its area of applications, and use the modified version to study the isomorphic classification of Cartesian products of the kind E0…a† E1 …b†; where a; b are sequences of positive numbers and p; q 2 ‰1; 1†: Let us note that in [19], [20] a complete isomorphic classification of these spaces is obtained in the case where at least one of the sequences a; b tends to 1 (i.e. at least one of the Cartesian factors is a Schwartz space). On the other hand, in the non-Schwartzian case a complete isomorphic classification of the spaces E0…a† E1 …b† is obtained in [2], [3] by using the appropriate linear topological invariants. In the same way one can characterize the isomorphisms of the spaces E0…a† E1 …b†; where p is fixed, p 2 ‰1; 1†: Here we complete these results by studying the non-Schwartzian case for p ˆj q: Some of our results are presented without proof in [4]. Ac knowl edg e m en t . We would like to thank professors S. L. Troyanski and V. P. Zahariuta for helpful discussions concerning the proof of Proposition 4. Preliminaries. Let X and Y be locally convex spaces and T : X ! Y be a continuous linear operator. The operator T is bounded (respectively precompact) if there exists a neighborhood U of zero in X such that T…U† is bounded (respectively precompact) in Y. The operator T is strictly singular if its restriction on any closed infinite-dimensional subspace of X is not an isomorphism. We write …X;Y† 2 b; …X;Y† 2k; …X;Y† 2ss; …X;Y† 2 bss if every continuous linear operator from X into Y is bounded, precompact, strictly singular, bounded and strictly singular, respectively. Since every precompact operator is bounded and strictly singular the relation …X;Y† 2k implies …X;Y† 2 bss: The converse is not true. For example, if 1 % p < q < 1 then …`p; `q† 2 bss, but …`p; `q†2jk since the identity mapping from `p to `q is not compact (see [7], Vol. I, Ch. 2. Sect. C). A KoÈ the matrix …aik† is of type …d1† or …d2†, respectively, if the following condition holds: 9 k0 8 k 9m; C : aik % Caik0 aim; …d1† 8 k 9m 8 ` 9C : Caim ^ aikai`: …d2† The corresponding KoÈ the spaces are referred as (d1) or (d2) spaces. It is easy to see that finite (respectively infinite) power series spaces are (d2) (respectively (d1)) spaces. V. P. Zahariuta [20] showed that …X;Y† 2 b if X and Y are locally convex spaces with absolute bases, satisfying the conditions …d2† and …d1† respectively. Of course then …X;Y† 2k if X is a Schwartz space or Y is a Montel space. D. Vogt [15] studied the relation …X;Y† 2 b for FreÂchet spaces. Using his results ( Satz 6.2 and Prop. 5.3 in [15]), one obtains that …X;Y† 2 b 58 P. B. DJAKOV, S. ÖNAL, T. TERZIOGÆ LU and M. YURDAKUL ARCH. MATH.