Skip to main content
Article thumbnail
Location of Repository

Banach Spaces of Analytic Vector-valued Functions

By Steven John Barclay


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\ud at the factorization of square matrix-valued boundary functions, beginning with spectral factorization in Chapter 2. Then ideas involving approximations with\ud 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\ud created by the noncommutativity of matrix multiplication.\ud \ud 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.\ud \ud 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\ud space equipped with a separable predual. We establish existence of a solution under mild assumptions, and characterise some of its properties. We also show\ud that in the scalar case, the unconstrained version of this problem is not well posed in general.\u

Publisher: School of Mathematics (Leeds)
Year: 2007
OAI identifier:

Suggested articles


  1. (1986). A problem of Douglas and Rudin on factorization, doi
  2. (1995). An introduction to infinite-dimensional linear systems theory, doi
  3. (2002). An introduction to measure and integration, 2nd ed.,
  4. (1998). AND OLIVI All., Weighted H2 approximation of transfer functions, doi
  5. (2003). Approximation in reflexive Banach spaces and applications to the invariant subspace problem,
  6. (2000). Approximation problems in some holomorphic spaces, with applications, Systems, approximation, singular integral operators, and related topics doi
  7. (1962). Barwach spaces of analytic functions, Prentice-Hall,
  8. (1981). Bounded Analytic Functions, doi
  9. (1990). C*-algebras and operator theory, doi
  10. Continuity of the spectral factorization mapping, doi
  11. (1999). Continuity of the spectral factorization on a vertical strip, doi
  12. (1985). Continuity of the spectral factorization operation,
  13. (2003). Hankel operators and their applications, Springer inonographs in mathematics, doi
  14. (1996). Hardy approxination to L°° functions on subsets of the circle, doi
  15. (1985). Hardy classes and operator theory, Oxford mathematical monographs, doi
  16. (1998). L1 factorization for Coo-contractions with isometric functional calculus, doi
  17. (1988). Linear operators, Part 1: general theory, Wiley classics library edition, Wiley-Interscience,
  18. (1963). Linear operators, Part 2: spectral theory,
  19. (1974). Monotone matrix functions and analytic continuation, doi
  20. (1976). On bandwidth, doi
  21. (2001). On the boundedness and continuity of the spectral factorization mapping, doi
  22. (2002). Operators, functions and systems: an easy reading,
  23. (2003). Parameter identification for Laplace equation and approximation in Hardy classes, doi
  24. (1958). Prediction theory and Fourier series in several variables, doi
  25. (1972). Some operator monotone functions, doi
  26. (1987). Spectral theory of self-adjoint operators in Hilbert space, doi
  27. (1977). Vector measures, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.