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Deformations of complete Pick spaces

Prahllad Deb, Jonathan Nureliyan, Eli Shamovich

Abstract

Motivated by the work of Pandey, Ofek, and Shalit on the one hand and deformation theory on the other, we study the Grassmannian of $n$-dimensional multiplier-coinvariant subspaces of the Drury-Arveson space. We show that this space admits a natural map to the symmetrized polyball that induces an isomorphism between the configuration space of $n$ points in the ball and the subspace of projection onto spaces spanned by $n$ distinct kernels. We discuss the tautological bundle on our Grassmannian and the corresponding operator algebra bundle. We construct examples of bundles of complete Pick spaces from homogeneous hypersurfaces in $\mathbb{B}_d$. Along with these bundles, we construct examples of Cowen-Doulas tuples of operators from the compressed Arveson $d$-shift.

Deformations of complete Pick spaces

Abstract

Motivated by the work of Pandey, Ofek, and Shalit on the one hand and deformation theory on the other, we study the Grassmannian of -dimensional multiplier-coinvariant subspaces of the Drury-Arveson space. We show that this space admits a natural map to the symmetrized polyball that induces an isomorphism between the configuration space of points in the ball and the subspace of projection onto spaces spanned by distinct kernels. We discuss the tautological bundle on our Grassmannian and the corresponding operator algebra bundle. We construct examples of bundles of complete Pick spaces from homogeneous hypersurfaces in . Along with these bundles, we construct examples of Cowen-Doulas tuples of operators from the compressed Arveson -shift.
Paper Structure (12 sections, 25 theorems, 79 equations)

This paper contains 12 sections, 25 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of the unit ball and its symmetrization
  4. Joint spectrum of commuting tuples
  5. Drury-Arveson space
  6. Finite-dimensional complete Pick spaces
  7. Multivariable Blaschke multipliers
  8. The Grassmannian of coinvariant spaces
  9. The subspace $\mathbb{P}_{d,n}^0$
  10. Bundles of complete Pick spaces and their multiplier algebras
  11. Cowen-Douglas class operators from hypersurfaces
  12. Stratification of $\mathrm{Pick}_n$

Key Result

Lemma 2.1

Let $0 < r < 1$, then for every two subsets of $n$ points $X, Y \subset r \overline{{\mathbb B}_d}$,

Theorems & Definitions (60)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 2.8
  • ...and 50 more