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

Abstract

Abstract. 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 E p 0 a  E q I b where 1 % pY q`I , p j q, a a n I n 1 and b b n I n 1 are sequences of positive numbers and E p 0 a , E q I b are respectively p -finite and q -infinite type power series spaces.

Nuclearity and multipliers between Banach spaces

“…In [7], quasidiagonal isomorphisms of l M -Köthe spaces and l N -Köthe spaces for different Orlicz functions M and N implies some kind of the "nuclearity" of the spaces (see 2.5). These two result are generalized in Theorem 2.7, and it is observed that quasidiagonal isomorphisms of different -Köthe spaces implies nuclearity which coincide with the multipliers of the corresponding spaces ∈ Λ and the results in [1] and [7] become the consequence of this theorem.…”

“…In [1], it is proved that, for different Köthe matrices A and B, quasidiagonal isomorphisms of l p -Köthe space and l q -Köthe space (1 ≤ p = q < ∞) implies "nuclearity" of the spaces (see 2.4). In [7], quasidiagonal isomorphisms of l M -Köthe spaces and l N -Köthe spaces for different Orlicz functions M and N implies some kind of the "nuclearity" of the spaces (see 2.5).…”

Remarks on strictly singular operators

“…Our starting point will be the class of Banach spaces. Applications of [7,Lemma 2] shall yield that some of these results are meaningful on the class of lcs's. In a particular case, to obtain a characterization (Theorem 3) is possible.…”

On Orlicz-Power Series Spaces

“…It is proved in [2] that the isomorphism of Köthe sequence spaces K lp (A) and K lq (B) implies the nuclearity of these spaces if p and q are different. This result was preceded by the result from [12] with the harder restriction K lp (A) = K lq (A).…”