## 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 [Tôhoku 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

### Citations:

Zbl 0361.47014; Zbl 0343.47032; Zbl 0471.47012
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