Kelly Bickel

Properties of Beurling-type submodules via Agler decompositions

Abstract In this paper, we study operator-theoretic properties of the compressed shift operators S z 1 and S z 2 on complements of submodules of the Hardy space over the bidisk H 2 ( D 2 ) . Specifically, we study Beurling-type submodules – namely submodules of the form θ H 2 ( D 2 ) for θ inner – using properties of Agler decompositions of θ to deduce properties of S z 1 and S z 2 on model spaces H 2 ( D 2 ) ⊖ θ H 2 ( D 2 ) . Results include characterizations (in terms of θ) of when a commutator [ S z j ⁎ , S z j ] has rank n and when subspaces associated to Agler decompositions are reducing for S z 1 and S z 2 . We include several open questions.

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

We analyze the singularities of rational inner functions (RIFs) on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative Hp membership purely in terms of contact order, a measure of the rate at which the zero set of an RIF approaches the distinguished boundary of the bidisk. We also show that derivatives of RIFs with singularities fail to be in Hp for p⩾32 and that higher non-tangential regularity of an RIF paradoxically reduces the Hp integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for Hp. Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of RIFs fails to belong to the unweighted Dirichlet space.

A multiplier algebra functional calculus

This paper generalizes the classical Sz.-Nagy--Foias

Bounds for the Hilbert Transform with Matrix A2 Weights

H^{\infty}(\mathbb{D})

Inner functions on the bidisk and associated Hilbert spaces

functional calculus for Hilbert space contractions. In particular, we replace the single contraction

A study of the matrix Carleson Embedding Theorem with applications to sparse operators

T

Canonical Agler decompositions and transfer function realizations

with a tuple

Two weight estimates with matrix measures for well localized operators

T=(T_1, \dots, T_d)

Characterizations of A2 matrix power weights

of commuting bounded operators on a Hilbert space and replace

Properties of vector-valued submodules on the bidisk

H^{\infty}(\mathbb{D})