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 arising from an inner function which is zero at zero by composition. The range of such an isometry is characterized as a closed subspace
of
(weak-
closed for
) satisfying the following: (i) the constant function 1 is in
; (ii) if
and
, then
; (iii) if
has inner-outer factorization
, then
is in
; (iv) if
is a collection of inner functions in
, then the greatest common divisor of
is also in
; and (v) if
, where
is inner and
, then
. 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
, 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.
- [1] Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 17. MR 27954, https://doi.org/10.1007/BF02395019
- [2] Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, https://doi.org/10.1007/978-1-4613-8208-9_19
- [3] James A. Deddens and Tin Kin Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261–273. MR 324467, https://doi.org/10.1090/S0002-9947-1973-0324467-0
- [4] Paul R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112. MR 152896, https://doi.org/10.1515/crll.1961.208.102
- [5] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
- [6] Michel Loève, Probability theory, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-New York-London, 1960. MR 0123342
- [7] Eric A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449. MR 223914, https://doi.org/10.4153/CJM-1968-040-4
- [8] Eric A. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175–181. MR 216321
- [9] Gian-Carlo Rota, On the representation of aeraging operators, Rend. Sem. Mat. Univ. Padova 30 (1960), 52–64. MR 112041
- [10] John V. Ryff, Subordinate 𝐻^{𝑝} functions, Duke Math. J. 33 (1966), 347–354. MR 192062
- [11] T. P. Srinivasan and Ju-kwei Wang, Weak *-Dirichlet algebras, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 216–249. MR 0198282
- [12] Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR 0275190
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1975-0358421-7
Article copyright:
© Copyright 1975
American Mathematical Society