Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Spectral theorem for unbounded strongly continuous groups on a Hilbert space


Authors: Khristo Boyadzhiev and Ralph deLaubenfels
Journal: Proc. Amer. Math. Soc. 120 (1994), 127-136
MSC: Primary 47D03; Secondary 47A60
DOI: https://doi.org/10.1090/S0002-9939-1994-1186983-0
MathSciNet review: 1186983
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ B$ is a closed, densely defined linear operator on a Hilbert space and $ a > 0$. Denote $ \{ z\vert\vert\operatorname{Im} (z)\vert < b\} $ by $ {H_b}$.

We show that $ B$ has an $ {H^\infty }({H_b})$ functional calculus, for all $ b > a$, if and only if $ iB$ generates a strongly continuous group of operators of exponential type $ a$. We obtain specific upper bounds on $ \vert\vert f(B)\vert\vert$, in terms of $ \sup \{ {e^{ - b\vert t\vert}}\vert\vert{e^{itB}}\vert\vert\;\vert t \in {\mathbf{R}}\} $.

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 $ {H^\infty }$ functional calculi.


References [Enhancements On Off] (What's this?)

  • [1] J. B. Baillon and Ph. Clement, Examples of unbounded imaginary powers of operators, J. Funct. Anal. 100 (1991), 419-434. 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, $ {H^\infty }$-functional calculus for perturbations of generators of holomorphic semigroups, Houston J. Math. 17 (1991), 131-147. MR 1107193 (92h:47052)
  • [4] -, Semigroups and resolvents of bounded variation, imaginary powers and $ {H^\infty }$ functional calculus, Semigroup Forum 45 (1992), 372-384. MR 1179859 (93i:47018)
  • [5] I. Cioranescu and L. Zsido, Analytic generators for one-parameter groups, Tôhoku Math. J. 28 (1976), 327-362. MR 0430867 (55:3872)
  • [6] M. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with an $ {H^\infty }$ functional calculus, in preparation.
  • [7] E. B. Davies, One-parameter 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), 189-201. MR 910825 (88m:47072)
  • [10] -, Some results about complex powers of closed operators, J. Math. Anal. Appl. 149 (1990), 124-136. MR 1054798 (91e:47038)
  • [11] X. T. Duong, $ {H^\infty }$ functional calculus of elliptic operators with $ {C^\infty }$ coefficients on $ {L^p}$ spaces of smooth domains, J. Austral. Math. Soc. Ser. A 48 (1990), 113-123. MR 1026842 (91c:47104)
  • [12] -, $ {H_\infty }$ functional calculus of second order elliptic $ PDE$ on $ {L^p}$ spaces, Miniconference on Operators in Analysis, Proc. Center Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1989, pp. 91-102.
  • [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), 289-309. MR 833395 (87i:47056)
  • [15] A. McIntosh, Operators which have an $ {H^\infty }$ functional calculus, Miniconference on Operator Theory and PDE, Proc. Center Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210-231. MR 912940 (88k:47019)
  • [16] A. McIntosh and A. Yagi, Operators of type $ \omega $ without a bounded $ {H^\infty }$-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), 429-452. MR 1038710 (91b:47030)
  • [18] W. Ricker, Spectral properties of the Laplace operator in $ {L^p}({\mathbf{R}})$, Osaka J. Math. 25 (1988), 399-410. MR 957870 (89h:47071)
  • [19] A. Yagi, Applications of the purely imaginary powers of operators in Hilbert spaces, J. Funct. Anal. 73 (1987), 216-231. MR 890664 (88g:47087)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47D03, 47A60

Retrieve articles in all journals with MSC: 47D03, 47A60


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1186983-0
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society