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A perturbation characterization of compactness of self-adjoint operators

Heydar Radjavi, Ping-Kwan Tam, Kok-Keong Tan (2003)

Studia Mathematica

A characterization of compactness of a given self-adjoint bounded operator A on a separable infinite-dimensional Hilbert space is established in terms of the spectrum of perturbations. An example is presented to show that without separability, the perturbation condition, which is always necessary, is not sufficient. For non-separable spaces, another condition on the self-adjoint operator A, which is necessary and sufficient for the perturbation, is given.

A q-analogue of complete monotonicity

Anna Kula (2008)

Colloquium Mathematicae

The aim of this paper is to give a q-analogue for complete monotonicity. We apply a classical characterization of Hausdorff moment sequences in terms of positive definiteness and complete monotonicity, adapted to the q-situation. The method due to Maserick and Szafraniec that does not need moments turns out to be useful. A definition of a q-moment sequence appears as a by-product.

A quasi-affine transform of an unbounded operator

Schôichi Ôta (1995)

Studia Mathematica

Some results on quasi-affinity for bounded operators are extended to unbounded ones and normal extensions of an unbounded operator are discussed in connection with quasi-affinity.

A remark concerning Putinar's model of hyponormal weighted shifts

Vasile Lauric (2018)

Czechoslovak Mathematical Journal

The question whether a hyponormal weighted shift with trace class self-commutator is the compression modulo the Hilbert-Schmidt class of a normal operator, remains open. It is natural to ask whether Putinar's construction through which he proved that hyponormal operators are subscalar operators provides the answer to the above question. We show that the normal extension provided by Putinar's theory does not lead to the extension that would provide a positive answer to the question.

A remark on extrapolation of rearrangement operators on dyadic H s , 0 < s ≤ 1

Stefan Geiss, Paul F. X. Müller, Veronika Pillwein (2005)

Studia Mathematica

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator T s , 0 < s < 2, to be the linear extension of the map ( h I ) / ( | I | 1 / s ) ↦ ( h τ ( I ) ) ( | τ ( I ) | 1 / s ) , where h I denotes the L ∞ -normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that T s ₀ is bounded on H s ₀ , then for all 0 < s < 2 the operator T s is bounded on H s .

A remark on Fefferman-Stein's inequalities.

Y. Rakotondratsimba (1998)

Collectanea Mathematica

It is proved that, for some reverse doubling weight functions, the related operator which appears in the Fefferman Stein's inequality can be taken smaller than those operators for which such an inequality is known to be true.

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