Hardy space expectation operators and reducing subspaces

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.