A semigroup approach to the numerical range of operators on Banach spaces

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• 作者：Martin Adler ; Waed Dada ; Agnes Radl
• 关键词：Numerical range ; Spectrum ; \(C_0\) ; semigroup
• 刊名：Semigroup Forum
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：94
• 期：1
• 页码：51-70
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algebra;
• 出版者：Springer US
• ISSN：1432-2137
• 卷排序：94

文摘

We introduce the numerical spectrum\(\sigma _n(A)\subseteq {\mathbb {C}}\) of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, \(\sigma _n(A)\) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of \(C_0\)-semigroups.