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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
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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.


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