Infinite-dimensional features of matrices and pseudospectra

TL;DR

This paper explores the properties of pseudospectra of Hilbert space operators, demonstrating how infinite-dimensional models can approximate finite matrices and produce diverse pseudospectral behaviors, including for nilpotent matrices.

Contribution

It establishes the existence of finite matrices approximating infinite-dimensional pseudospectra and uses operator models to generate examples with complex pseudospectral properties.

Findings

Finite matrices can approximate the pseudospectra of operators uniformly.

Pseudospectra of nilpotent matrices can be arbitrarily complex.

The function \\|\\sqrt{S-z}\\| can oscillate rapidly far from the spectrum.

Abstract

Given a Hilbert space operator $T$, the level sets of function $\Psi_T(z)=\|(T-z)^{-1}\|^{-1}$ determine the so-called pseudospectra of $T$. We set $\Psi_T$ to be zero on the spectrum of $T$. After giving some elementary properties of $\Psi_T$ (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator $T$, there is a sequence $\{T_n\}$ of finite matrices such that $\Psi_{T_n}(z)$ tends to $\Psi_{T}(z)$ uniformly on $\C$. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices. One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large order. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen--Douglas class operators. We use these results to show that, to the opposite to the function $\Psi_{S}$, the function $\|\sqrt{S-z}\,\|$ for certain finite matrices $S$ may oscillate arbitrarily fast even far away from the spectrum.