Spectral theory for contraction semigroups on Hilbert space

Gearhart, Larry

doi:10.1090/S0002-9947-1978-0461206-1

Spectral theory for contraction semigroups on Hilbert space
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by Larry Gearhart PDF

Trans. Amer. Math. Soc. 236 (1978), 385-394 Request permission

Abstract:

In this paper we determine the relationship between the spectra of a continuous contraction semigroup on Hilbert space and properties of the resolvent of its infinitesimal generator. The methods rely heavily on dilation theory. In particular, we reduce the general problem to the case that the cogenerator of the semigroup has a characteristic function with unitary boundary values. We then complete the analysis by generalizing the scalar result of J. W. Moeller on compressions of the translation semigroup to the case of infinite multiplicity.

References

R. G. Douglas, P. S. Muhly, and Carl Pearcy, Lifting commuting operators, Michigan Math. J. 15 (1968), 385–395. MR 236752, DOI 10.1307/mmj/1029000093
Paul A. Fuhrmann, On the corona theorem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc. 132 (1968), 55–66. MR 222701, DOI 10.1090/S0002-9947-1968-0222701-7
Paul A. Fuhrmann, A functional calculus in Hilbert spaces based on operator valued analytic functions, Israel J. Math. 6 (1968), 267–278. MR 236735, DOI 10.1007/BF02760259

On the spectral theory of the translation semigroup and its commutant

Paul R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112. MR 152896, DOI 10.1515/crll.1961.208.102
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Peter D. Lax, Translation invariant spaces, Acta Math. 101 (1959), 163–178. MR 105620, DOI 10.1007/BF02559553
Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
James W. Moeller, On the spectra of some translation invariant spaces, J. Math. Anal. Appl. 4 (1962), 276–296. MR 150592, DOI 10.1016/0022-247X(62)90055-0
J. W. Moeller, Translation invariant spaces with zero-free spectra, Duke Math. J. 31 (1964), 99–108. MR 167837
R. S. Phillips, Spectral theory for semi-groups of linear operators, Trans. Amer. Math. Soc. 71 (1951), 393–415. MR 44737, DOI 10.1090/S0002-9947-1951-0044737-9
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Béla Sz.-Nagy and Ciprian Foiaş, On the structure of intertwining operators, Acta Sci. Math. (Szeged) 35 (1973), 225–254. MR 399896