MULTIPLIERS FOR OPERATOR-VALUED BESSEL SEQUENCES AND GENERALIZED HILBERT-SCHMIDT CLASSES

Abstract

In 1960, Schatten studied operators of the form Σ∞ <inf>n=1</inf> λ<inf>n</inf>(x<inf>n</inf>⊗y<inf>n</inf>), where {x<inf>n</inf>}<inf>n</inf> and {y<inf>n</inf>}<inf>n</inf> are orthonormal sequences in a Hilbert space, and {λ<inf>n</inf>}<inf>n</inf> ∈ ℓ∞(ℕ). Balazs generalized some of the results of Schatten in 2007. In this paper, we further generalize results of Balazs by studying the operators of the form Σ∞ <inf>n=1</inf> λ<inf>n</inf>(A∗ <inf>n</inf>x<inf>n</inf> ⊗ B∗ <inf>n</inf> y<inf>n</inf>), where {A<inf>n</inf>}<inf>n</inf> and {B<inf>n</inf>}<inf>n</inf> are operator-valued Bessel sequences, {x<inf>n</inf>}<inf>n</inf> and {y<inf>n</inf>}<inf>n</inf> are sequences in the Hilbert space such that {∥x<inf>n</inf>∥∥y<inf>n</inf>∥}<inf>n</inf> ∈ ℓ∞(ℕ). We also generalize the class of Hilbert-Schmidt operators studied by Balazs. © 2022 KSCAM.