標題: Numerical ranges of nilpotent operators 作者: Gau, Hwa-LongWu, Pei Yuan應用數學系Department of Applied Mathematics 關鍵字: numerical range;numerical radius;nilpotent operator 公開日期: 1-Aug-2008 摘要: For any operator A on a Hilbert space, let W(A), w(A) and w(0)(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 A(n) = 0, then w(A) <= (n - 1)w(0)(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A) = (n - 1)w(0)(A), (2)A is unitarily equivalent to an operator of the form aA(n) circle plus A', where a is a scalar satisfying vertical bar a vertical bar = 2w(0)(A), A(n) is the n-by-n matrix [GRAPHICS] and A' is some other operator, and (3) W(A) = bW(A(n)) for some scalar b. (C) 2008 Elsevier Inc. All rights reserved. URI: http://dx.doi.org/10.1016/j.laa.2008.03.029http://hdl.handle.net/11536/8506 ISSN: 0024-3795 DOI: 10.1016/j.laa.2008.03.029 期刊: LINEAR ALGEBRA AND ITS APPLICATIONS Volume: 429 Issue: 4 起始頁: 716 結束頁: 726 Appears in Collections: Articles

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