Automatic continuity for weakly decomposable operators

Abstract

The authors introduce a new class of operators that are relative to the identity, and some of their properties are derived; for example, these operators have the single valued extension property. The main result is that every generalized intertwining of an operator having property (δ) with such a one is necessarily bounded whenever certain side conditions are satisfied. Examples also show that this class of operators is not comparable by inclusion to the classical cases (e.g. operators). In this note we shall generalize some results of Laursen and Neumann [13] on the automatic continuity of intertwinings of operators with certain spectral decomposition properties. Specifically, we prove that a linear map which is a generalized intertwining of an operator satisfying property (δ) with a second operator that is relative to the identity (WDI) is necessarily continuous (bounded) (provided the operator pair has no critical eigenvalue). The properties (δ) and WDI are both relaxations of the notion decomposable but in different directions; see below for details. We mention that some of our results and proofs have been improved and shortened by appeals to [13], which appeared after the original submission of the present paper. Section 2 of the paper deals with definitions and other background needed for this study. Some results are proved which have their own independent interest. For example, we show that the notions of admissible and subadmissibleoperator (see [13, 15]) are identical. Our Proposition 2.1 seems to be known, but the authors do not know where it may have been published. In Section 3 we give the definition of our new weakly operators and then establish some of their elementary properties, among which is the single-valued extension property. We show through two examples that, unlike other subclasses with spectral decomposition, our new class is not comparable by inclusion to the class of operators.