Numerical ranges of nilpotent operators

Hwa Long Gau; Pei Yuan Wu

doi:10.1016/j.laa.2008.03.029

Numerical ranges of nilpotent operators

Hwa Long Gau
, Pei Yuan Wu

Department of Mathematics

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

15 Scopus citations

Abstract

For any operator A on a Hilbert space, let W (A), w (A) and w₀ (A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if Aⁿ = 0, then w (A) ≤ (n - 1) w₀ (A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w (A) = (n - 1) w₀ (A), (2) A is unitarily equivalent to an operator of the form aA_n ⊕ A^′, where a is a scalar satisfying | a | = 2 w₀ (A), A_n is the n-by-n matrixfenced((0, 1, ⋯, 1; 0, {triple dot, diagonal NW-SE}, ⋮; {triple dot, diagonal NW-SE}, 1; 0))andA^′ is some other operator, and (3) W (A) = bW (A_n) for some scalar b.

Original language	English
Pages (from-to)	716-726
Number of pages	11
Journal	Linear Algebra and Its Applications
Volume	429
Issue number	4
DOIs	https://doi.org/10.1016/j.laa.2008.03.029
State	Published - 1 Aug 2008

Keywords

Nilpotent operator
Numerical radius
Numerical range

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title = "Numerical ranges of nilpotent operators",

abstract = "For any operator A on a Hilbert space, let W (A), w (A) and w0 (A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if An = 0, then w (A) ≤ (n - 1) w0 (A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w (A) = (n - 1) w0 (A), (2) A is unitarily equivalent to an operator of the form aAn ⊕ A′, where a is a scalar satisfying | a | = 2 w0 (A), An is the n-by-n matrixfenced((0, 1, ⋯, 1; 0, \{triple dot, diagonal NW-SE\}, ⋮; \{triple dot, diagonal NW-SE\}, 1; 0))andA′ is some other operator, and (3) W (A) = bW (An) for some scalar b.",

keywords = "Nilpotent operator, Numerical radius, Numerical range",

author = "Gau, \{Hwa Long\} and Wu, \{Pei Yuan\}",

note = "Funding Information: ∗ Corresponding author. E-mail addresses: [email protected] (H.-L. Gau), [email protected] (P.Y. Wu). 1 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-008-006. 2 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-009-013-MY3 and by the MOE-ATU.",

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doi = "10.1016/j.laa.2008.03.029",

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T1 - Numerical ranges of nilpotent operators

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information: ∗ Corresponding author. E-mail addresses: [email protected] (H.-L. Gau), [email protected] (P.Y. Wu). 1 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-008-006. 2 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-009-013-MY3 and by the MOE-ATU.

PY - 2008/8/1

Y1 - 2008/8/1

N2 - For any operator A on a Hilbert space, let W (A), w (A) and w0 (A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if An = 0, then w (A) ≤ (n - 1) w0 (A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w (A) = (n - 1) w0 (A), (2) A is unitarily equivalent to an operator of the form aAn ⊕ A′, where a is a scalar satisfying | a | = 2 w0 (A), An is the n-by-n matrixfenced((0, 1, ⋯, 1; 0, {triple dot, diagonal NW-SE}, ⋮; {triple dot, diagonal NW-SE}, 1; 0))andA′ is some other operator, and (3) W (A) = bW (An) for some scalar b.

AB - For any operator A on a Hilbert space, let W (A), w (A) and w0 (A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if An = 0, then w (A) ≤ (n - 1) w0 (A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w (A) = (n - 1) w0 (A), (2) A is unitarily equivalent to an operator of the form aAn ⊕ A′, where a is a scalar satisfying | a | = 2 w0 (A), An is the n-by-n matrixfenced((0, 1, ⋯, 1; 0, {triple dot, diagonal NW-SE}, ⋮; {triple dot, diagonal NW-SE}, 1; 0))andA′ is some other operator, and (3) W (A) = bW (An) for some scalar b.

KW - Nilpotent operator

KW - Numerical radius

KW - Numerical range

UR - https://www.scopus.com/pages/publications/44649195624

U2 - 10.1016/j.laa.2008.03.029

DO - 10.1016/j.laa.2008.03.029

M3 - 期刊論文

AN - SCOPUS:44649195624

SN - 0024-3795

VL - 429

SP - 716

EP - 726

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 4

ER -

Numerical ranges of nilpotent operators

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