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We prove that if X is a Banach space, R, S ∈ B(X), then RS is subscalar (subdecomposable) if and only if SR is. As corollaries, it is shown that w-hyponormal operators (including p-hyponormal (p > 0) and log-hyponormal operators) and their Aluthge transformations and inverse Aluthge transformations are subscalar. (2000). Primary 47B99; Secondary 47A10. Mathematics Subject Classification

The paper gives characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball Bn in C n .

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence (f n ) ∞ n=1 so that f n , x n → 0 as n → ∞ for every weakly null sequence (x n ) ∞ n=1 in X; X has property (µ s ) if every weak * null sequence in X * admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property (µ s ) and reflexivity (or even the Grothendieck property) imply property (K). In the present paper we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

This paper shows that every w*-lower semicontinuous Lipschitzian convex function on the dual of a locally uniformly convexifiable Banach space, in particular, the dual of a separable Banach space, can be uniformly approximated by a generically Fréchet differentiable w*-lower semicontinuous monotone-nondecreasing Lipschitzian convex function sequence © 2002 Elsevier Science (USA)