Multirectangular Characteristic Invariants for Power L-Kothe Spaces of First Type

Abstract

Let l be a Banach sequence space with a monotone norm \\ center dot \\(l), in which the canonical system (e(i)) is a normalized unconditional basis. We consider the problem of quasi-diagonal isomorphism of first type power l-Kothe spaces E-l (lambda, a) (see (1) below). From [P.A. Chalov, V.P. Zahariuta, On quasi-diagonal isomorphism of generalized power spaces, in: Linear Topological Spaces and Complex Analysis, vol. 2, METU - TUBITAK, Ankara, 1995, pp. 35-44; P.A. Chalov, T. Terzioglu, V.P. Zahariuta, First type power Kothe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3, METU - TUBITAK, Ankara, 1997, pp. 30-44; P.A. Chalov, T. Terzioglu, V.P. Zahariuta, Multirectangular invariants for power Kothe spaces, J. Math. Anal. Appl. 297 (2004) 673-695] it is known that the system of all m-rectangle characteristics mu(m) (see (9) below) is a complete quasi-diagonal invariant on the class of all first type power Kothe spaces [V.P. Zahariuta, On isomorphisms and quasi-equivalence of bases of power Kothe spaces, Soviet Math. Dokl. 16 (1975) 411-414; V.P. Zahariuta, Linear topologic invariants and their applications to isomorphic classification of generalized power spaces, Turkish J. Math. 20 (1996) 237-289], if the relation of equivalency of systems (mu(X)(m)) and (mu((X) over tilde)(m)) is defined by some natural estimates with constants independent of m. Deriving the characteristic (beta) over tilde from the characteristic beta (see [V.P. Zahariuta, Linear topological invariants and isomorphisms of spaces of analytic functions, in: Matem. Analiz i ego Pril., vol. 2, Rostov Univ., Rostov-on-Don, 1970, pp. 3-13 (in Russian), in: Matem. Analiz i ego Pril., vol. 3, Rostov Univ., Rostov-on-Don, 1971, pp. 176-180 (in Russian); V.P. Zahariuta, Generalized Mityagin invariants and a continuum of mutually nonisomorphic spaces of analytic functions, Funktsional. Anal. i Prilozhen. 11 (1977) 24-30 (in Russian); V.P. Zahariuta, Compact operators and isomorphisms of Kothe spaces, in: Aktualnye Voprosy Matem. Analiza, vol. 46, Rostov Univ., Rostov-on-Don, 1978, pp. 62-71 (in Russian); P.A. Chalov, P.B. Djakov, V.P. Zahariuta, Compound invariants and embeddings of Cartesian products, Studia Math. 137 (1) (1999) 33-47; P.B. Djakov, M. Yurdakul, V.P. Zahariuta, Isomorphic classification of Cartesian products, Michigan Math. J. 43 (1996) 221-229; V.P. Zahariuta, Linear topologic invariants and their applications to isomorphic classification of generalized power spaces, Turkish J. Math. 20 (1996) 237-289], and using the S. Krein's interpolation method of analytic scale, we are able to generalize some results of [P.A. Chalov, V.P. Zahariuta, On quasi-diagonal isomorphism of generalized power spaces, in: Linear Topological Spaces and Complex Analysis, vol. 2, METU - TUBITAK, Ankara, 1995, pp. 35-44; P.A. Chalov, T. Terzioglu, V.P. Zahariuta, First type power Kothe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3, METU - TUBITAK, Ankara, 1997, pp. 30-44; P.A. Chalov, T. Terzioglu, V.P. Zahariuta, Multirectangular invariants for power Kothe spaces, J. Math. Anal. Appl. 297 (2004) 673-695]. (C) 2007 Elsevier Inc. All rights reserved.