Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T04:29:03.682Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiplication Invariant Subspaces of Hardy Spaces

Published online by Cambridge University Press:  20 November 2018

T. L. Lance  and
M. I. Stessin
T. L. Lance
Affiliation:
Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY, USA 12222 e-mail: [email protected], [email protected]
M. I. Stessin
Affiliation:
Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY, USA 12222 e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies closed subspaces L of the Hardy spaces Hp which are g-invariant (i.e., g. L ⊆ L) where g is inner, g ≠ 1. If p = 2, theWold decomposition theorem implies that there is a countable “g-basis” f1, f2, . . . of L in the sense that L is a direct sum of spaces fj . H2[g] where H2[g] = {f o g | f ∈ H2}. The basis elements fj satisfy the additional property that ∫T |fj|2gk = 0, k = 1, 2, . . . . We call such functions g-2-inner. It also follows that any f ∈ H2 can be factored f = hf ,2 . (F2 o g) where hf,2 is g-2-inner and F is outer, generalizing the classical Riesz factorization. Using Lp estimates for the canonical decomposition of H2,we find a factorization f = hf ,p.(Fpog) for f ∈ Hp. If p ≤ 1 and g is a finite Blaschke product we obtain, for any g-invariant L ⊆ Hp, a finite g-basis of g-p-inner functions.

Keywords

30H0546E1547B38
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 100 - 118
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Arveson, W.B., Subalgebras of C*-algebras, Acta Math. 123(1969), 141–224.Google Scholar
2. Cima, J.A., MacGregor, T.H., Stessin, M.I., Recapturing functions in Hp, Indiana University Math. J. 41(1994), 523–533.Google Scholar
3. Doob, J.L., Measure Theory, Springer-Verlag, New York, 1994.Google Scholar
4. Foias, C., Nagy, B.Sz., Harmonic Analysis of Operators on Hilbert Space, North Holland, 1970.Google Scholar
5. Garnett, J.B., Bounded Analytic Functions, Acad. Press, New York, 1981.Google Scholar
6. Halmos, P.R., Shifts on Hilbert Spaces, J. Reine Angew. Math. 208(1961), 102–112.Google Scholar
7. Lancaster, P., The Theory of Matrices With Applications, Acad. Press, New York, 1985.Google Scholar
8. Mascioni, V., Ideals of the disc algebra, operators related to Hilbert space contractions and complete boundedness, Houston J. Math. (2). 20(1994), 299–311.Google Scholar
9. Paulsen, V.I., Every completely polynomialy bounded operator is similar to a contraction, J. Funct. Anal. 55(1984), 1–17.Google Scholar
10. Stessin, M.I., Invariant minimal interpolation in spaces of analytic functions, preprint.Google Scholar