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A note on the commutator of two operators on a locally convex space

Edvard Kramar (2016)

Commentationes Mathematicae Universitatis Carolinae

Denote by C the commutator A B - B A of two bounded operators A and B acting on a locally convex topological vector space. If A C - C A = 0 , we show that C is a quasinilpotent operator and we prove that if A C - C A is a compact operator, then C is a Riesz operator.

A note on the convolution theorem for the Fourier transform

Charles S. Kahane (2011)

Czechoslovak Mathematical Journal

In this paper we characterize those bounded linear transformations T f carrying L 1 ( 1 ) into the space of bounded continuous functions on 1 , for which the convolution identity T ( f * g ) = T f · T g holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

A note on the differentiable structure of generalized idempotents

Esteban Andruchow, Gustavo Corach, Mostafa Mbekhta (2013)

Open Mathematics

For a fixed n > 2, we study the set Λ of generalized idempotents, which are operators satisfying T n+1 = T. Also the subsets Λ†, of operators such that T n−1 is the Moore-Penrose pseudo-inverse of T, and Λ*, of operators such that T n−1 = T* (known as generalized projections) are studied. The local smooth structure of these sets is examined.

A note on the powers of Cesàro bounded operators

Zoltán Léka (2010)

Czechoslovak Mathematical Journal

In this note we give a negative answer to Zem�nek’s question (1994) of whether it always holds that a Cesàro bounded operator T on a Hilbert space with a single spectrum satisfies lim n T n + 1 - T n = 0 .

A note on the range of generalized derivation.

Mohamed Amouch (2006)

Extracta Mathematicae

Let L(H) denote the algebra of bounded linear operators on a complex separable and infinite dimensional Hilbert space H. For A, B ∈ L(H), the generalized derivation δA,B associated with (A, B), is defined by δA,B(X) = AX - XB for X ∈ L(H). In this note we give some sufficient conditions for A and B under which the intersection between the closure of the range of δA,B respect to the given topology and the kernel of δA*,B* vanishes.

A note on γ-radonifying and summing operators

Zdzisław Brzeźniak, Hongwei Long (2015)

Banach Center Publications

In this note, we discuss certain generalizations of γ-radonifying operators and their applications to the regularity for linear stochastic evolution equations on some special Banach spaces. Furthermore, we also consider a more general class of operators, namely the so-called summing operators and discuss the application to the compactness of the heat semi-group between weighted L p -spaces.

A notion of analytic generator for groups of unbounded operators

José E. Galé (2005)

Banach Center Publications

We introduce a notion of analytic generator for groups of unbounded operators, on Banach modules, arising from Esterle’s quasimultiplier theory. Characterizations of analytic generators are given in terms of the existence of certain functional calculi. This extends recent results about C₀ groups of bounded operators. The theory is applicable to sectorial operators, representations of H , and integrated groups.

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