Banach Spaces of Analytic Vector-valued Functions

Abstract

The main theme of the thesis is the study of continuity and approximation problems, involving matrix-valued and vector-valued Hardy spaces on the unit disc ID and its boundary T in the complex plane. The first part of the thesis looks at the factorization of square matrix-valued boundary functions, beginning with spectral factorization in Chapter 2. Then ideas involving approximations with inner and outer functions are used to solve a matrix analogue of the Douglas-Rudin problem in Chapter 3. In both cases, considerable considerable extra difficulties are created by the noncommutativity of matrix multiplication.

More specifically, we show that the matrix spectral factorization mapping is sequentially continuous from LP to H2p (where 1<p< co), under the additional assumption of uniform integrability of the logarithms of the spectral densities to be factorized. We show, moreover, that this condition is necessary for continuity, as well as sufficient. Concerning the Douglas-Rudin problem in Chapter 3, we show that any log-integrable essentially bounded square matrix-valued function f can be written in the form h*g, where lt and g lie in H. Extensions to other LP spaces, with norni bounds on the factors of f, are also provided.

The final part of the thesis takes a somewhat different direction. In Chapter 4, we consider the problem of weighted H°° approximation of vector-valued L°° functions on the unit circle, subject to a weighted sup-type constraint on the size of the approximant. This involves the development of a suitable theory of vector-valued L°° and H°° functions on T, taking values in an arbitrary Banach space equipped with a separable predual. We establish existence of a solution under mild assumptions, and characterise some of its properties. We also show that in the scalar case, the unconstrained version of this problem is not well posed in general.