Let X, Y be Banach spaces, S: X → Y and R: Y → X be bounded operators. We investigate common spectral properties of RS and SR. We then apply the result obtained to extensions, Aluthge transforms and upper triangular operator matrices.
Let R and S be two operators on a Hilbert space. We discuss the link between the subscalarity of RS and SR. As an application, we show that backward Aluthge iterates of hyponormal operators and p-quasihyponormal operators are subscalar.
We establish a non-commutative analog of the classical Banach Principle on the almost everywhere convergence of sequences of measurable functions. The result is stated in terms of quasi-uniform (or almost uniform) convergence of sequences of measurable (with respect to a trace) operators affiliated with a semifinite von Neumann algebra. Then we discuss possible applications of this result.
Let X be a Banach space. Let 𝓐(X) be a closed ideal in the algebra ℒ(X) of the operators acting on X. We say that ℒ(X)/𝓐(X) is a Calkin algebra whenever the Fredholm operators on X coincide with the operators whose class in ℒ(X)/𝓐(X) is invertible. Among other examples, we have the cases in which 𝓐(X) is the ideal of compact, strictly singular, strictly cosingular and inessential operators, and some other ideals introduced as perturbation classes in Fredholm theory. Our aim is to present some...
Using axiomatic joint spectra we obtain a functional calculus which extends our previous Gelfand-Waelbroeck type results to include a Banach-valued Taylor-Waelbroeck spectrum.
In this paper we consider Bessel equations of the type , where A is an nn complex matrix and X(t) is an nm matrix for t > 0. Following the ideas of the scalar case we introduce the concept of a fundamental set of solutions for the above equation expressed in terms of the data dimension. This concept allows us to give an explicit closed form solution of initial and two-point boundary value problems related to the Bessel equation.
We determine the norm in , 1 < p < ∞, of the operator , where and are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...
Let be a Banach space and be a bounded linear operator on . We denote by the set of all complex such that does not have the single-valued extension property at . In this note we prove equality up to between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl’s theorem for operator matrices and multiplier operators.