Some convexity theorems for the generalized numerical ranges
1996; Taylor & Francis; Volume: 40; Issue: 3 Linguagem: Inglês
10.1080/03081089608818441
ISSN1563-5139
Autores Tópico(s)Advanced Optimization Algorithms Research
ResumoAbstract Let Mn be the algebra n × n complex matrices, where n ≥ 2. Given a nonscalar matrix C ∊ Mn , the C-numerical range of A ∊ Mn is defined by If rank (C − γI) for some γ ∊ C(which is always true when n = 2) then W(C:A) be written as a + bW (q:A) for some a,b ∊ C and q ∊ C and q ∊[0,1], where is the q-numerical range of A. We give short proofs for the faets that W(q :A) is convex for all A ∊ Mn , and that W(C : A) is an elliptical disk if A,C ∊ M2 . These results have been proved by Tsing and Nakazato, respectively, by some very involved computational methods. 1Research partially supported by a NATO grant. This paper was done while the author was visiting the Mathematics Department of the University of Hong Kong. He would like to think the staff of the department for their warm hosnitality. 1Research partially supported by a NATO grant. This paper was done while the author was visiting the Mathematics Department of the University of Hong Kong. He would like to think the staff of the department for their warm hosnitality. Notes 1Research partially supported by a NATO grant. This paper was done while the author was visiting the Mathematics Department of the University of Hong Kong. He would like to think the staff of the department for their warm hosnitality.
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