On the directional growth of the resolvent norm
Horia Cornean, Henrik Garde, Arne Jensen
TL;DR
This work addresses how the resolvent norm $\|R_A(z)\|$ can grow directionally when moving from a point $z$ in the resolvent set, by introducing a norm-determining pair $(z,\{\psi_n\})$ and expanding the resolvent via iterated identities. Depending on three sets of assumptions, the authors prove explicit linear or quadratic growth along a short line segment $[z,z']$ contained in the resolvent set, with $z'$ chosen in a specific near-direction determined by $\theta_0$ and a small $a_0>0$. The results yield criteria for the absence of interior points in level sets of $\|R_A(\cdot)\|$ and for the existence of local minima, and they are illustrated through normal and non-normal operator examples. An elementary finite-dimensional application shows that every point in an $\varepsilon$-pseudospectrum can be connected to an eigenvalue by a polygonal path within the pseudospectrum, highlighting practical implications for pseudospectral geometry. Overall, the paper provides a concrete, constructive framework to analyze how resolvent-norm growth shapes pseudospectra for both infinite- and finite-dimensional operators.
Abstract
Let $A$ be a densely defined closed operator on a separable Hilbert space $\mathcal{H}$. Its resolvent is denoted by $R_A(z)=(A-zI)^{-1}$. Under very general assumptions at the point $z$, we construct a line segment $[z,z']$ in the resolvent set on which the resolvent norm grows. We obtain estimates of the type $||R_A(ζ)|| \geq ||R_A(z)|| + C|ζ-z|^δ$ for $ζ\in[z,z']$ with either $δ=1$ or $δ=2$. We give a number of examples involving both normal and non-normal operators, and an application to pseudospectra of matrices.