Functional Commutativity and Spectral Representation of Analytic Operator-valued Functions that Commute with their Derivatives
by
Daniel Turcotte
Ryerson Polytechnic University

Goff showed in 1981 that a self-adjoint analytic family of matrices that commutes with its derivative is functionally commutative. We generalize this result in two ways. Firstly, we study analytic families of operators in a Banach space that commute with their derivatives and give a detailed description of their spectral properties, their representations and their subspaces of eigenvectors in terms of analytic eigenfibers and the constant projections that are associated with them. In particular, we consider the case of an analytic family (A(s))_{s in \eusmD} of operators in \eusmL(\eusbX) that commutes with its derivative and for which there is an uncountable subset \eusmG of \eusmD such that, for each s in \eusmG, A(s) is a compact spectral operator of scalar type, where \eusbX is a complex Banach space and \eusmD a region of the complex plane, among other things we show that the family (A(s))_{s in \eusmD} is then functionally commutative, that A(s) is a spectral operator compact of scalar type for all s in \eusmD and we give the spectral representation of (A(s))_{s in \eusmD}. We also study these families of operators in a Hilbert space \eusbH where we introduce and apply the theory of propagation of involutive properties of the space \eusmL(\eusbH) of bounded linear operators on \eusbH, and we give the relationship that exists with normal compact operators. In particular, we generalize the Spectral Theorem for a Compact Normal Operator to the case of an analytic family (A(s))_{s in I} of compact normal operators that commutes with its derivative.

Secondly, by applying these results to case of matrices, we generalize Goff's theorem in the particular case of matrices.

The first paper on this subject was published in 1950 by Ascoli, and was concerned with families of matrices. Since that first paper, all the publications remained concerned with families of matrices [Schwerdtfeger, 1952], [Dieudonné, 1974], [Goff, 1981] (submitted by Olga Taussky Todd), [Evard, 1985]). This is the first paper where analytic families of operators on infinite-dimensional Banach and Hilbert spaces that commute with their derivatives are treated. We have obtained an extensive theory of these analytic families of operators that commute with their derivatives. A very short selection of the results will covered.

Date received: May 1, 1999