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The biholomorphic invariance of essential normality on bounded symmetric domains

Lijia Ding

TL;DR

The paper addresses the problem of biholomorphic invariance for $p$-essential normality of Hilbert modules on bounded symmetric domains by developing two main invariance results: Theorem 1, which uses new integral formulas for the Taylor functional calculus to show $p$-essential normality is preserved under permissive linear transformations and automorphisms, and Theorem 2, which establishes intrinsic biholomorphic invariance of $p$-essential normality for quotient submodules under automorphism multipliers. It further analyzes the Wallach set case, computes the Taylor spectrum for compression tuples, and derives corona solvability for quotient submodules, with applications to the Geometric Arveson-Douglas conjecture and hyperrigidity. The results tie together operator-theoretic invariants with complex-geometric structures, providing coordinate-chart independence and enabling a geometric formulation of key conjectures. Overall, the work extends invariant theory for $p$-essential normality to broad classes of domains and submodules, linking dilation theory, index theory, and complex geometry.

Abstract

This paper mainly concerns the biholomorphic invariance of $p$-essential normality of Hilbert modules on bounded symmetric domains. By establishing new integral formulas concerning rational function kernels for the Taylor functional calculus, we prove a biholomorphic invariance result related to the $p$-essential normality. Furthermore, for quotient analytic Hilbert submodules determined by analytic varieties, we develop an algebraic approach to proving that the $p$-essential normality is preserved invariant if the coordinate multipliers are replaced by arbitrary automorphism multipliers. Moreover, the Taylor spectrum of the compression tuple is calculated under a mild condition, which gives a solvability result of the corona problem for quotient submodules. As applications, we extend the recent results on the equivalence between $\infty$-essential normality and hyperrigidity.

The biholomorphic invariance of essential normality on bounded symmetric domains

TL;DR

The paper addresses the problem of biholomorphic invariance for -essential normality of Hilbert modules on bounded symmetric domains by developing two main invariance results: Theorem 1, which uses new integral formulas for the Taylor functional calculus to show -essential normality is preserved under permissive linear transformations and automorphisms, and Theorem 2, which establishes intrinsic biholomorphic invariance of -essential normality for quotient submodules under automorphism multipliers. It further analyzes the Wallach set case, computes the Taylor spectrum for compression tuples, and derives corona solvability for quotient submodules, with applications to the Geometric Arveson-Douglas conjecture and hyperrigidity. The results tie together operator-theoretic invariants with complex-geometric structures, providing coordinate-chart independence and enabling a geometric formulation of key conjectures. Overall, the work extends invariant theory for -essential normality to broad classes of domains and submodules, linking dilation theory, index theory, and complex geometry.

Abstract

This paper mainly concerns the biholomorphic invariance of -essential normality of Hilbert modules on bounded symmetric domains. By establishing new integral formulas concerning rational function kernels for the Taylor functional calculus, we prove a biholomorphic invariance result related to the -essential normality. Furthermore, for quotient analytic Hilbert submodules determined by analytic varieties, we develop an algebraic approach to proving that the -essential normality is preserved invariant if the coordinate multipliers are replaced by arbitrary automorphism multipliers. Moreover, the Taylor spectrum of the compression tuple is calculated under a mild condition, which gives a solvability result of the corona problem for quotient submodules. As applications, we extend the recent results on the equivalence between -essential normality and hyperrigidity.
Paper Structure (9 sections, 23 theorems, 136 equations)

This paper contains 9 sections, 23 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. The proof of Theorem 1
  4. The proof of Theorem 2
  5. The case on the Wallach set
  6. The Taylor spectrum of the compression
  7. applications
  8. Geometric Arveson-Douglas conjecture
  9. Hyperrigidity and $\infty$-essential normality

Key Result

Lemma 3.1

Let $U, V, W$ be bounded operators on a Banach space, then the following hold:

Theorems & Definitions (51)

  • Example 2.1
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 41 more