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2-summing multiplication operators

Dumitru Popa (2013)

Studia Mathematica

Let 1 ≤ p < ∞, = ( X ₙ ) n ∈ ℕ be a sequence of Banach spaces and l p ( ) the coresponding vector valued sequence space. Let = ( X ₙ ) n ∈ ℕ , = ( Y ₙ ) n ∈ ℕ be two sequences of Banach spaces, = ( V ₙ ) n ∈ ℕ , Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator M : l p ( ) → l q ( ) by M ( ( x ₙ ) n ∈ ℕ ) : = ( V ₙ ( x ₙ ) ) n ∈ ℕ . We give necessary and sufficient conditions for M to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.

A counterexample in operator theory

Antonio Córdoba (1993)

Publicacions Matemàtiques

The purpose of this note is to give an explicit construction of a bounded operator T acting on the space L2[0,1] such that |Tf(x)| ≤ ∫01 |f(y)| dy for a.e. x ∈ [0.1], and, nevertheless, ||T||Sp = ∞ for every p &lt; 2. Here || ||Sp denotes the norm associated to the Schatten-Von Neumann classes.

A remark on the range of elementary operators

Said Bouali, Youssef Bouhafsi (2010)

Czechoslovak Mathematical Journal

Let L ( H ) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L ( H ) , we define the elementary operator Δ A : L ( H ) ⟶ L ( H ) by Δ A ( X ) = A X A - X . In this paper we study the class of operators A ∈ L ( H ) which have the following property: A T A = T implies A T * A = T * for all trace class operators T ∈ C 1 ( H ) . Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ A is closed under taking...

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