TY - JOUR
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 - http://www.scopus.com/inward/record.url?scp=44649195624&partnerID=8YFLogxK
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 -