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On the analytical generator of a group of operators. (English) Zbl 0608.47048

The analytical generator B of a group of operators \(\{U(t)|\) \(t\in {\mathbb{R}}\}\) was introduced by I. Cioranescu and L. Zsido [TĂ´hoku Math. J., II. Ser. 28, 327-362 (1976; Zbl 0361.47014) and Rev. Roum. Math. Pur. Appl. 21, 817-850 (1976; Zbl 0343.47032)] to examine some spectral properties of the infinitesimal generator iT. If the group is non-quasianalytic, then \[ {\check \mu}(T)x:=\int U(t)x d\mu (t), \] \({\check \mu}\) the Fourier transform of the measure \(\mu\), is a functional calculus in the sense of Colojoara and Foias. This calculus was extended by the author to a large algebra of local multipliers in [Math. Ann. 260, 143-150 (1982; Zbl 0471.47012)] and thus \(B=\exp \{T\}\). This proves the spectral properties of B, in particular, the various definitions of spectral subspaces of T, B, and U(\(\cdot)\) are identical. The ”spectrum problem” is that of determining whether (a) \(\sigma (B)=\exp \sigma (T)\) or (b) \(\sigma (B)={\hat {\mathbb{C}}}\). We reprove a lot of conditions equivalent to (a) and that (a) holds iff B is decomposable. The inequality \(\| \hat f\|_ 1\leq (\delta /\pi)^{1/r}\| f\|_ r+(\delta^{1-s}/[\pi (s-1)])^{1/s}\| f'\|_ s\), \(\delta >0\), \(1\leq r\leq 2\), \(1<s\leq 2\), yields good estimations of the quasispectral projection \(\| P_{r,\epsilon}\|\) and T is ”well- bounded” iff \(\| P_{r,\epsilon}\| \leq K\) for \(r\in {\mathbb{R}}\), \(\epsilon >0\). Finally there are simple examples which distinguish the different cases.

MSC:

47D03 Groups and semigroups of linear operators
47A10 Spectrum, resolvent
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47A60 Functional calculus for linear operators
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