Continuity Properties of the Spectrum of Operators on Lebesgue Spaces

Abstract

Abstract.Fix 1 < p < s < oo . Let Tx,x e [p,s], be the collection of bounded linear operators on the Lebesgue spaces Lx determined by some fixed operator T . This paper concerns continuity properties of the map x -* a(Tx).

Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

“…In the recent literature, one sometimes says that A is inverse closed in B [3,4] or that A is a spectral subalgebra of B [26,28].…”

“…In a sense they extend our results for locally compact groups [15] to Banach algebras of operators which have much less structure. In this regard, Barnes's results on Banach algebras of integral operators [3] need to be mentioned. These are in a similar spirit, but unfortunately they work only for sublinear weights.…”

Uniform Error Estimates of Operational Quadrature Methods for Nonlinear Convolution Equations on the Half-Line

“…Even the simple case when rf = {Si} is a singleton is not clear , but if the class f is essentially closed under adjugation, then the case p = X can be handled as well; see §8. Perhaps this is the right time to mention the work of Barnes [1]. He considers the class of integral operators Si, (3.3), which (only) have the property that for some a > 0, (3.9) ess sup / (l + |í-T|n|zc(í,T)| + |zc(T,Z)|}í/T is bounded for zz = 0, and tends to zero as zz -► oo.…”

“…For equations of the above type (1.1), there is a well-developed theory in an L2 setting. The Circle Condition Theorem (Theorem 2.1 below) says that if there exists a circle in the complex plane such that || takes all its values inside the circle and -X/K(s) takes all its values outside of it for Re s > 0, then we have an L estimate (1)(2)(3)(4)(5) \\x -x\\L2{&+) < c||/-/||¿2(R+)…”

Wiener's Lemma for Infinite Matrices of Gohberg-Baskakov-Sjöstrand Class

“…Schur class is not inverse-closed in B(ℓ 2 ) but the weighted Gröchenig-Schur class is when the weight satisfies the GRS-condition [2,6,7,9,17,19,22]. The Gohberg-Baskakov-Sjöstrand class [3,19,20].…”