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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
47D03 Groups and semigroups of linear operators
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