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On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space

Driss Drissi (1998)

Studia Mathematica

Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.

On a Kleinecke-Shirokov theorem

Vasile Lauric (2021)

Czechoslovak Mathematical Journal

We prove that for normal operators N 1 , N 2 ( ) , the generalized commutator [ N 1 , N 2 ; X ] approaches zero when [ N 1 , N 2 ; [ N 1 , N 2 ; X ] ] tends to zero in the norm of the Schatten-von Neumann class 𝒞 p with p > 1 and X varies in a bounded set of such a class.

On a theorem of Gelfand and its local generalizations

Driss Drissi (1997)

Studia Mathematica

In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, T 1 , . . . , T m , on a Banach space X.

On a Weyl-von Neumann type theorem for antilinear self-adjoint operators

Santtu Ruotsalainen (2012)

Studia Mathematica

Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl-von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint operator is the sum of a diagonalizable operator and of a compact operator with arbitrarily small Schatten p-norm. On the way, we discuss conjugations and their properties. A spectral integral representation for antilinear self-adjoint operators is constructed....

On an inclusion between operator ideals

Manuel A. Fugarolas (2011)

Czechoslovak Mathematical Journal

Let 1 q < p < and 1 / r : = 1 / p max ( q / 2 , 1 ) . We prove that r , p ( c ) , the ideal of operators of Geľfand type l r , p , is contained in the ideal Π p , q of ( p , q ) -absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.

On bounded approximation properties of Banach spaces

Eve Oja (2010)

Banach Center Publications

This survey features some recent developments concerning the bounded approximation property in Banach spaces. As a central theme, we discuss the weak bounded approximation property and the approximation property which is bounded for a Banach operator ideal. We also include an overview around the related long-standing open problem: Is the approximation property of a dual Banach space always metric?

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