Abstract
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.
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Notes
For every \(x\in X\), the duality set \({\mathfrak {J}}(x)\) is defined as
We call \(\sigma _r(A):=\{\lambda \in {{\mathbb {C}}}: rg(\lambda -A)\text { is not dense in }X\}\) the residual spectrum of A. It coincides with the point spectrum \(\sigma _p(A^\prime )\) of the adjoint \(A'\) of A, see [6, Prop. IV.1.12].
For \( 0<\delta <\frac{\pi }{2}\) and \(\theta \in [0,2\pi )\), \(z\in {{\mathbb {C}}}\) we define \(\Sigma ^{\theta }_{\delta }+z:=\{ z+ e^{-i\theta }\;\lambda \in {{\mathbb {C}}}: |arg \lambda |\le \delta \}\).
In the following, \({\mathbb {C}}_{-}\) denotes the closed left half plane, i.e., \({\mathbb {C}}_{-}:=\{\lambda \in {\mathbb {C}}:\hbox {Re}\,{\lambda }\le 0\}\).
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The authors thank Prof. Rainer Nagel for many valuable discussions and remarks.
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Communicated by Abdelaziz Rhandi.
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Adler, M., Dada, W. & Radl, A. A semigroup approach to the numerical range of operators on Banach spaces. Semigroup Forum 94, 51–70 (2017). https://doi.org/10.1007/s00233-015-9752-y
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DOI: https://doi.org/10.1007/s00233-015-9752-y