The Single-Valued Extension Property Is Not Preserved Under Sums and Products of Commuting Operators

Abstract

Abstract. We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.2000 Mathematics Subject Classification. Primary 47B37. Secondary 47A10, 47A11. Introduction.The problem of studying whether or not the local spectral properties such as the single-valued extension property, Dunford's condition (C), Bishop's property (β), the decomposition property (δ), or decomposability are preserved under sums and products of commuting operators has been considered by several authors, and remains an open problem; see for instance [3], [6], [9] and the references therein. Partial positive answers were obtained but only in certain special cases. In [9], T. L. Miller and M. M. Neumann showed that the sum and the product of two commuting operators with Dunford's condition (C) have the single-valued extension property. They also proved that the product of two commuting operators has this property provided that one of them is non-invertible and has fat local spectra.In this note, we show that the single-valued extension property is not preserved, in general, under the sums and products of commuting operators, and prove that the product of two commuting operators has this property provided that the intersection of their analytic cores is trivial. Our counter-examples are provided by tensor products of backward and forward unilateral weighted shifts, and our arguments and ideas are influenced by the ones given in [9].We now gather together some basic facts about the single-valued extension property and local spectrum. Our reference is the excellent book of K. Laursen and M. M. Neumann [6].Let X be a complex Banach space, and let L(X) be the algebra of bounded linear operators on X. For an operator T ∈ L(X), we denote as usual its spectrum and its

On the stability of the localized single-valued extension property under commuting perturbations

“…While a specific example based on the theory of weighted shifts may be found in [11], here we present a general principle that shows that such examples exist in abundance. While a specific example based on the theory of weighted shifts may be found in [11], here we present a general principle that shows that such examples exist in abundance.…”