Unitary rho-dilations and the Holbrook radius for bounded operators on Hilbert space

Abstract

In this thesis we deal with the theory of unitary p-dilations of bounded operators on a Hilbert space H, as developed by Sz. Nagy and Foias, and a related functional on B(H), the algebra of bounded linear operators on H. In the first Chapter we consider the classes of operators possessing a unitary p-dilation, and obtain their basic properties , using an approach which adapts itself to a unified treatment. Next in Chapter 2, we examine the behaviour of the sequence of powers of an element p arbitrary and positive, and we show that the sequence converges to a non-negative limit, less than or equal to this is a generalization of a result by M.J. Crabb in which he considers the special case p=2. We then give an intrinsic characterization of the elements x in and obtain various results concerning the structure of operators which satisfy for some For every , the classes , turn out to be balanced, absorbing sets of operators which contain the zero operator, and hence a generalized Minkowski functional may be unambiguously defined on them by This functional, usually referred to in the literature as the Holbrook radius of T, plays a very important role in the study of unitary p-dilations, since the elements T of are characterized by [diagram]. The basic properties of the Holbrook radius for a bounded operator are studied in Chapter 3. A number of new results concerning the Holbrook radius of nilpotent operators of arbitrary index greater than 2 are obtained which enable us to have a clearer view of the general structure of the [mathematical symbol][rho] classes, in a unified framework.