Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Hardy space expectation operators and reducing subspaces


Author: Joseph A. Ball
Journal: Proc. Amer. Math. Soc. 47 (1975), 351-357
DOI: https://doi.org/10.1090/S0002-9939-1975-0358421-7
MathSciNet review: 0358421
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: In this paper we study the range of the isometry on $ {H^p}$ arising from an inner function which is zero at zero by composition. The range of such an isometry is characterized as a closed subspace $ \mathfrak{M}$ of $ {H^p}$ (weak-$ ^ \ast $ closed for $ p = \infty $) satisfying the following: (i) the constant function 1 is in $ \mathfrak{M}$; (ii) if $ f \in \mathfrak{M}$ and $ g \in {H^\infty } \cap \mathfrak{M}$, then $ fg \in \mathfrak{M}$; (iii) if $ f \in \mathfrak{M}$ has inner-outer factorization $ f = \chi \cdot F$, then $ \chi $ is in $ \mathfrak{M}$; (iv) if $ \{ {B_\alpha }:\alpha \in \mathcal{A}\} $ is a collection of inner functions in $ \mathfrak{M}$, then the greatest common divisor of $ \{ {B_\alpha }:\alpha \in \mathcal{A}\} $ is also in $ \mathfrak{M}$; and (v) if $ f \in \mathfrak{M},B \in \mathfrak{M}$, where $ B$ is inner and $ \bar B \cdot f \in {H^p}$, then $ \bar B \cdot f \in \mathfrak{M}$. The proof makes use of the fact that there exists a projection onto such a subspace satisfying the axioms of an expectation operator, which for $ p = 2$, is simply the orthogonal projection. This characterization is applied to give an equivalent formulation of a conjecture of Nordgren concerning reducing subspaces of analytic Toeplitz operators.


References [Enhancements On Off] (What's this?)

  • [1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 17pp. MR 10, 381. MR 0027954 (10:381e)
  • [2] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963), 89-102. MR 28 #3350. MR 0160136 (28:3350)
  • [3] J. A. Deddens and Tin Kin Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973) , 261-273. MR 0324467 (48:2819)
  • [4] P. R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102-112. MR 27 #2868. MR 0152896 (27:2868)
  • [5] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood. Cliffs, N. J., 1965. MR 0133008 (24:A2844)
  • [6] M. Loève, Probability theory, 2nd ed., University Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 23 #A670. MR 0123342 (23:A670)
  • [7] E. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. MR 36 #6961. MR 0223914 (36:6961)
  • [8] -, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175-181. MR 35 #7155. MR 0216321 (35:7155)
  • [9] G. C. Rota, On the representation of averaging operators, Rend. Sem. Mat. Univ. Padova 30 (1960), 52-64. MR 22 #2899. MR 0112041 (22:2899)
  • [10] J. Ryff, Subordinate $ {H^p}$ functions, Duke Math. J., 33 (1966), 347-354. MR 33 #289. MR 0192062 (33:289)
  • [11] T. P. Srinivasan and Ju-Kwei Wang, Weak*-Dirichlet algebras, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965), edited by F. Virtel, Scott, Foresman, Chicago, Ill., 1966, pp. 216-249. MR 33 #6441. MR 0198282 (33:6441)
  • [12] B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, Akad. Kiadó, Budapest, 1970; English transl., North-Holland, Amsterdam; American Elsevier, New York, 1971. MR 43 #947. MR 0275190 (43:947)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0358421-7
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society