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Fackler, Stephan

A short counterexample to the inverse generator problem on non-Hilbertian reflexive \(L^p\)-spaces. (English) Zbl 1359.47038

Arch. Math. 106, No. 4, 383-389 (2016).
Let \(e^{tA} \) be a strongly continuous \(C_{0} \)-semigroup of Banach space operators. Suppose that its generator \(A\) is one-to-one and has dense range. Its inverse \(A^{-1} \) is also densely defined, closed, and with dense range. It is interesting to know whether \(A^{-1} \) also generates a \(C_{0} \) semigroup. The question is important, because in the case when the semigroup \(e^{tA} \) is analytical and bounded on some sector, the answer is positive, as \(A^{-1} \) is sectorial in the same sector.
A counterexample is known for the problem on \(L^{p} \) spaces with \(p>1\), \(p\neq 2\). The author constructs here a different counterexample for the same case. The short proof is based on Fourier analysis. As the author notes, the inverse generator problem remains open for semigroups on Hilbert spaces.
Reviewer: Khristo N. Boyadzhiev (Ada)

Cited in 3 Documents

MSC:

47D06 One-parameter semigroups and linear evolution equations

Keywords:

inverse generator; transference principle
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\textit{S. Fackler}, Arch. Math. 106, No. 4, 383--389 (2016; Zbl 1359.47038)
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