A local directional growth estimate of the resolvent norm

Research output: Working paperResearch

Abstract

We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.
Original languageEnglish
PublisherarXiv.org
Number of pages16
Publication statusPublished - 2018

Fingerprint

Resolvent
Norm
Estimate
Closed Operator
Unbounded Operators
Maximum Principle
Linear Operator
Hilbert space
Operator
Class

Cite this

@techreport{818bec0f68df4cb1aae041f7a1ecce53,
title = "A local directional growth estimate of the resolvent norm",
abstract = "We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.",
author = "Cornean, {Decebal Horia} and Henrik Garde and Arne Jensen and Kn{\"o}rr, {Hans Konrad}",
year = "2018",
language = "English",
publisher = "arXiv.org",
type = "WorkingPaper",
institution = "arXiv.org",

}

TY - UNPB

T1 - A local directional growth estimate of the resolvent norm

AU - Cornean, Decebal Horia

AU - Garde, Henrik

AU - Jensen, Arne

AU - Knörr, Hans Konrad

PY - 2018

Y1 - 2018

N2 - We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.

AB - We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.

M3 - Working paper

BT - A local directional growth estimate of the resolvent norm

PB - arXiv.org

ER -