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.
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 -spaces.
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 , and integrated groups.
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.
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.
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.
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.
The uniqueness of the Wold decomposition of a finite-dimensional stationary process without assumption of full rank stationary process and the Lebesgue decomposition of its spectral measure is easily obtained.
For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator , 0 < s < 2, to be the linear extension of the map , where denotes the -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 is bounded on , then for all 0 < s < 2 the operator is bounded on .