The biholomorphic invariance of essential normality on bounded symmetric domains

TL;DR

This paper investigates the invariance of $p$-essential normality of Hilbert modules on bounded symmetric domains under biholomorphic maps, introducing new integral formulas and algebraic methods to establish invariance and related spectral properties.

Contribution

It establishes the biholomorphic invariance of $p$-essential normality using new integral formulas and algebraic techniques, extending previous results and analyzing spectral properties.

Findings

Biholomorphic invariance of $p$-essential normality proved.

New integral formulas for rational function kernels developed.

Taylor spectrum of the compression tuple calculated under mild conditions.

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.