On the normalized numerical range

Ilya M. Spitkovsky; Andrei Florian Stoica

doi:10.7153/oam-11-15

On the normalized numerical range

Ilya M. Spitkovsky, Andrei Florian Stoica

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The normalized numerical range of an operator A is defined as the set F_N(A) of all the values 〈Ax, x〉/||Ax|| attained by unit vectors x ∉ ker A. We prove that F_N(A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shifts, isometries, essentially Hermitian) the complete description of F_N(A) is obtained.

Original language	English (US)
Article number	11-15
Pages (from-to)	219-240
Number of pages	22
Journal	Operators and Matrices
Volume	11
Issue number	1
DOIs	https://doi.org/10.7153/oam-11-15
State	Published - Mar 2017

Keywords

Essentially Hermitian operator
Normalized numerical range
Numerical range
Partial isometry
Weighted shift

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

Access to Document

10.7153/oam-11-15

Cite this

@article{0e496a2ce4f246c9aa5749982fb970f3,

title = "On the normalized numerical range",

abstract = "The normalized numerical range of an operator A is defined as the set FN(A) of all the values 〈Ax, x〉/||Ax|| attained by unit vectors x ∉ ker A. We prove that FN(A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shifts, isometries, essentially Hermitian) the complete description of FN(A) is obtained.",

keywords = "Essentially Hermitian operator, Normalized numerical range, Numerical range, Partial isometry, Weighted shift",

author = "Spitkovsky, {Ilya M.} and Stoica, {Andrei Florian}",

note = "Publisher Copyright: {\textcopyright} ELEMENT, Zagreb.",

year = "2017",

month = mar,

doi = "10.7153/oam-11-15",

language = "English (US)",

volume = "11",

pages = "219--240",

journal = "Operators and Matrices",

issn = "1846-3886",

publisher = "Element d.o.o.",

number = "1",

}

TY - JOUR

T1 - On the normalized numerical range

AU - Spitkovsky, Ilya M.

AU - Stoica, Andrei Florian

N1 - Publisher Copyright: © ELEMENT, Zagreb.

PY - 2017/3

Y1 - 2017/3

N2 - The normalized numerical range of an operator A is defined as the set FN(A) of all the values 〈Ax, x〉/||Ax|| attained by unit vectors x ∉ ker A. We prove that FN(A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shifts, isometries, essentially Hermitian) the complete description of FN(A) is obtained.

AB - The normalized numerical range of an operator A is defined as the set FN(A) of all the values 〈Ax, x〉/||Ax|| attained by unit vectors x ∉ ker A. We prove that FN(A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shifts, isometries, essentially Hermitian) the complete description of FN(A) is obtained.

KW - Essentially Hermitian operator

KW - Normalized numerical range

KW - Numerical range

KW - Partial isometry

KW - Weighted shift

UR - http://www.scopus.com/inward/record.url?scp=85013634219&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013634219&partnerID=8YFLogxK

U2 - 10.7153/oam-11-15

DO - 10.7153/oam-11-15

M3 - Article

AN - SCOPUS:85013634219

SN - 1846-3886

VL - 11

SP - 219

EP - 240

JO - Operators and Matrices

JF - Operators and Matrices

IS - 1

M1 - 11-15

ER -

On the normalized numerical range

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this