Inner Functions in Reproducing Kernel Spaces

Abstract

In Beurlings approach to inner functions for the shift operator S on the Hardy space H 2 , a function f is inner when f ⊥ S n f for all n 1. Inspired by this approach, this paper develops a notion of an inner vector x for any operator T on a Hilbert space, via the analogous condition x ⊥ T n x for all n 1. We study these inner vectors in a variety of settings. Using Birkhoff-James orthogonality, we extend this notion of inner vector for an operator on a Banach space. We then apply this development of inner function to recast a theorem of Shapiro and Shields to discuss the zero sets for functions in Hilbert spaces, as well as obtain a corresponding result for zero sets for a wide class of Banach spaces.

On the geometry of the multiplier space of $\ell_A^p$

“…Apart from a harmless multiplicative constant, this definition is equivalent to the traditional meaning of "inner" when p = 2. Furthermore, this approach to defining an inner property is consistent with that taken in other function spaces [1,3,15,17,20,21,22,31,32].…”

Optimal approximants and orthogonal polynomials in several variables

“…We say that g ∈ H(Ω) \ {0} is weakly inner if g, χ j g = 0 for all j = 0. See [13] for a comprehensive overview of notions of inner function for a wide range of reproducing kernel Hilbert spaces. Inner functions can also be defined for Banach spaces of analytic functions in a similar fashion using the notion of Birkhoff-James orthogonality, viz.…”

“…Inner functions can also be defined for Banach spaces of analytic functions in a similar fashion using the notion of Birkhoff-James orthogonality, viz. [13,Section 7]. Proposition 7.…”

“…Shields [39], we can build weakly inner functions in any reproducing kernel Hilbert space with a finite prescribed zero set. See [13] and [28] for further generalizations.…”

Inner vectors for Toeplitz operators

“…In this paper we survey, continue, and synthesize some discussions begun in [4,10,11] dealing with the notion of an "inner vector" for T ϕ along with the general notion of an inner vector for a contraction on a Hilbert space. We connect these results with the operator-valued Poisson kernel and some work from [2,3] concerning "factoring an L 1 function through a contraction".…”