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A spectral mapping theorem for representations of one-parameter groups. (English) Zbl 1131.47041

Let \(\omega\) be a weight on \(\mathbb R\) (i.e., \(\omega(t)\geq 1\) and \(\omega(t+s)\leq\omega(t)\omega(s)\)). Denote by \(M_\omega(\mathbb R)\) the convolution algebra of all complex measures \(\mu\) on \(\mathbb R\) such that \(\|\mu\|=\int_{\mathbb R}\omega(t)| \mu| (t)<\infty\), and by \(\operatorname{reg} M_\omega(\mathbb R)\) its greatest regular subalgebra.
Let \(X\) be a Banach space and \({\mathbf T}=\{T(t)\}_{t\in\mathbb R}\subset B(X)\) a \(C_0\)-group such that \(\|T(t)\|\leq\omega(t)\), \(t\in\mathbb R\), where \(\omega\) satisfies \(\int\frac{\omega(t)}{1+t^2 }\,dt<\infty\). The author’s main result states that if \(A\) is the generator of \({\mathbf T}\) (which can be unbounded), then
\[ \sigma(\widehat\mu({\mathbf T}))=\overline {\widehat\mu(\sigma (iA))} \]
for all \(\mu\in\operatorname{reg}M_\omega(\mathbb R)\), where \(\widehat\mu\) denotes the Fourier–Stieltjes transform.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47A10 Spectrum, resolvent
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46J05 General theory of commutative topological algebras
22D20 Representations of group algebras
22D15 Group algebras of locally compact groups
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