Abstract :
Let be a Banach sequence space with a monotone norm · , in which the canonical system (ei ) is
a normalized unconditional basis. We consider the problem of quasi-diagonal isomorphism of first type
power -Köthe spaces E (λ, 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 – TÜB˙ITAK, Ankara, 1995, pp. 35–44; P.A. Chalov, T. Terzio˘glu, V.P. Zahariuta, First type power
Köthe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3,
METU – TÜB˙ITAK, Ankara, 1997, pp. 30–44; P.A. Chalov, T. Terzio˘glu, V.P. Zahariuta, Multirectangular
invariants for power Köthe spaces, J. Math. Anal. Appl. 297 (2004) 673–695] it is known that the system
of all m-rectangle characteristics μm (see (9) below) is a complete quasi-diagonal invariant on the class
of all first type power Köthe spaces [V.P. Zahariuta, On isomorphisms and quasi-equivalence of bases of
power Köthe 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 (μX
m) and (μ X
m) is defined by some natural estimates
with constants independent of m. Deriving the characteristic β˜ from the characteristic β (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 Köthe
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 clas-sification 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 – TÜB˙ITAK, Ankara, 1995,
pp. 35–44; P.A. Chalov, T. Terzio˘glu, V.P. Zahariuta, First type power Köthe spaces and m-rectangular invariants,
in: Linear Topological Spaces and Complex Analysis, vol. 3, METU – TÜB˙ITAK, Ankara, 1997,
pp. 30–44; P.A. Chalov, T. Terzio˘glu, V.P. Zahariuta, Multirectangular invariants for power Köthe spaces,
J. Math. Anal. Appl. 297 (2004) 673–695].
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