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On a question of T. Kato. (Russian) Zbl 0545.47025
The author has proved that the Kato hypothesis [see T. Kato; Arch. Rat. Mech. Anal. 10, 273-275 (1962; Zbl 0106.311)] is false, but after some correction of assumptions the following result is true. Let E be a Banach space and U(t,A) a semigroup of linear operators in E of class $$C_ 0$$ with the generator A and $$\omega_ A:=\lim_{t\to 0}t^{- 1}\ln \| U(t,A)\|.$$ Let A, B, C be the generators of semigroups of class $$C_ 0$$ and $$E_ 1$$ (resp. $$E_ 2)$$ the completion of a subspace $$D\subset E$$ with the norm $$\| u\|_ 1=\| u\| +\| Bu\| +\| Cu\|$$ (resp.$$\| u\|_ 2=\| u\| +\| Cu\|).$$ If A,B,C satisfy the Schrödinger relation $$[A,B]u=ABu-BAu=Cu, [A,C]u=[B,C]u=0$$ for $$u\in D$$, D is dense in E and there exist a,b,$$c\in {\mathbb{C}}$$ such that $$Re a>\omega_ A, Re b>\omega_ B, Re c>\omega_ C$$ and $$(aI-A)D, (bI-B)D$$, $$(cI-C)D$$ are dense respectively in $$E_ 1,E_ 2$$ and E then the Weyl relation $$U(t,A)U(s,B)=U(ts,C)U(s,B)U(t,A),$$ t,$$s\geq 0$$ holds.
Reviewer: M.C.Zdun
##### MSC:
 47D03 Groups and semigroups of linear operators
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