Two counterexamples in semigroup theory on Hilbert space

Chernoff, Paul R.

doi:10.1090/S0002-9939-1976-0399952-4

Two counterexamples in semigroup theory on Hilbert space
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by Paul R. Chernoff

Proc. Amer. Math. Soc. 56 (1976), 253-255

DOI: https://doi.org/10.1090/S0002-9939-1976-0399952-4

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Abstract:

There exist $({C_0})$ semigroups ${T_1}(t),{T_2}(t)$ on Hilbert space with the following properties: ${T_1}$ has a bounded generator and is uniformly bounded, but is not similar to a contraction semigroup. ${T_2}$ is uniformly bounded, and there exists no scalar $\alpha$ such that ${e^{ - \alpha t}}{T_2}(t)$ is similar to a contraction semigroup.

References

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523

Contraction semigroups on Hilbert space

21

A problem concerning semigroups on Hilbert space

Heinz-Otto Kreiss, Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand. 7 (1959), 71–80 (German). MR 110952, DOI 10.7146/math.scand.a-10563
Edward W. Packel, A semigroup analogue of Foguel’s counterexample, Proc. Amer. Math. Soc. 21 (1969), 240–244. MR 238124, DOI 10.1090/S0002-9939-1969-0238124-7

Bibliographic Information

Journal: Proc. Amer. Math. Soc. 56 (1976), 253-255
DOI: https://doi.org/10.1090/S0002-9939-1976-0399952-4
MathSciNet review: 0399952