Spectral theorem for unbounded strongly continuous groups on a Hilbert space
Authors:
Khristo Boyadzhiev and Ralph deLaubenfels
Journal:
Proc. Amer. Math. Soc. 120 (1994), 127136
MSC:
Primary 47D03; Secondary 47A60
MathSciNet review:
1186983
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Abstract 
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Abstract: Suppose is a closed, densely defined linear operator on a Hilbert space and . Denote by . We show that has an functional calculus, for all , if and only if generates a strongly continuous group of operators of exponential type . We obtain specific upper bounds on , in terms of . Corollaries include the spectral theorem for closed operators on a Hilbert space and a generalization of a result due to McIntosh relating imaginary powers and functional calculi.
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 J. B. Baillon and Ph. Clement, Examples of unbounded imaginary powers of operators, J. Funct. Anal. 100 (1991), 419434. MR 1125234 (92j:47036)
 [2]
 M. Balabane, H. Emamirad, and M. Jazar, Spectral distributions and generalization of Stone's theorem, Math. Z. (to appear).
 [3]
 K. Boyadzhiev and R. deLaubenfels, functional calculus for perturbations of generators of holomorphic semigroups, Houston J. Math. 17 (1991), 131147. MR 1107193 (92h:47052)
 [4]
 , Semigroups and resolvents of bounded variation, imaginary powers and functional calculus, Semigroup Forum 45 (1992), 372384. MR 1179859 (93i:47018)
 [5]
 I. Cioranescu and L. Zsido, Analytic generators for oneparameter groups, Tôhoku Math. J. 28 (1976), 327362. MR 0430867 (55:3872)
 [6]
 M. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with an functional calculus, in preparation.
 [7]
 E. B. Davies, Oneparameter semigroups, Academic Press, London, 1980. MR 591851 (82i:47060)
 [8]
 R. deLaubenfels, Unbounded holomorphic functional calculus and abstract Cauchy problems for operators with polynomially bounded resolvents, J. Funct. Anal. (to appear). MR 1223706 (94h:47029)
 [9]
 G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189201. MR 910825 (88m:47072)
 [10]
 , Some results about complex powers of closed operators, J. Math. Anal. Appl. 149 (1990), 124136. MR 1054798 (91e:47038)
 [11]
 X. T. Duong, functional calculus of elliptic operators with coefficients on spaces of smooth domains, J. Austral. Math. Soc. Ser. A 48 (1990), 113123. MR 1026842 (91c:47104)
 [12]
 , functional calculus of second order elliptic on spaces, Miniconference on Operators in Analysis, Proc. Center Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1989, pp. 91102.
 [13]
 J. A. Goldstein, Semigroups of operators and applications, Oxford, New York, 1985.
 [14]
 E. Marschall, On the analytic generator of a group of operators, Indiana Univ. Math. J. 35 (1986), 289309. MR 833395 (87i:47056)
 [15]
 A. McIntosh, Operators which have an functional calculus, Miniconference on Operator Theory and PDE, Proc. Center Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210231. MR 912940 (88k:47019)
 [16]
 A. McIntosh and A. Yagi, Operators of type without a bounded functional calculus, Miniconference on Operators in Analysis, Proc. Center Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1989.
 [17]
 J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), 429452. MR 1038710 (91b:47030)
 [18]
 W. Ricker, Spectral properties of the Laplace operator in , Osaka J. Math. 25 (1988), 399410. MR 957870 (89h:47071)
 [19]
 A. Yagi, Applications of the purely imaginary powers of operators in Hilbert spaces, J. Funct. Anal. 73 (1987), 216231. MR 890664 (88g:47087)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199411869830
PII:
S 00029939(1994)11869830
Article copyright:
© Copyright 1994
American Mathematical Society
