Reflexivity, Algebraic Reflexivity and Linear Interpolation

Abstract

Introduction. This paper was motivated by a discovery that most finite dimensional subalgebras of (B(H), for H an infinite dimensional Hilbert space, are reflexive, with the only obstructions to reflexivity being finite-rank considerations. Proofs (section 2) do not depend on multiplicative structure, nor on topology, so extend to linear subspaces of transformations in an abstract setting. Abstract reflexivity has been studied in [3, 4, 7], primarily for singly generated algebras. Reflexivity properties can be interpreted as linear interpolation properties, and we shall adopt this point of view. The results of section 2 extend, in section 3, to algebraic reflexivity counterparts for countably generated (algebraically) linear subspaces of bounded linear transformations acting on a Banach space. As consequences, in section 4 we obtain generalizations of two single operator results. One is a multivariate version of the result, due to Kaplansky [10, Theorem 15], that a bounded locally algebraic operator acting on a Banach space is algebraic. The other is a multivariate version of the result, due to Douglas and Foias [5] for Hilbert space and to Hadwin [7] for a general Banach space, extending work of Fillmore [6], that a bounded non-algebraic operator acting on a Banach space is (topologically) algebraically reflexive. In section 5 we give an application, kindly suggested by D. Hadwin, concerning joint strong similarity orbits of n-tupules of operators. We wish to thank E. Azoff, K. Davidson, J. Erdos, D. Hadwin, J. Kraus, C. Lance and A. Sourour for conversations, and some correspon-