Some remarks on quasi-analytic vectors

Chernoff, Paul R.

doi:10.1090/S0002-9947-1972-0295125-5

Some remarks on quasi-analytic vectors
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by Paul R. Chernoff PDF

Trans. Amer. Math. Soc. 167 (1972), 105-113 Request permission

Abstract:

Recently a number of authors have developed conditions of a generalized quasi-analytic nature which imply essential selfadjointness for semibounded symmetric operators in Hilbert space. We give a unified derivation of these results by reducing them to the basic theorems of Nelson and Nussbaum. In addition we present an extension of Nussbaum’s quasi-analytic vector theorem to the setting of semigroups in Banach spaces.

References

Jerome A. Goldstein, Semigroups and second-order differential equations, J. Functional Analysis 4 (1969), 50–70. MR 0254668, DOI 10.1016/0022-1236(69)90021-4
Minoru Hasegawa, On quasi-analytic vectors for dissipative operators, Proc. Amer. Math. Soc. 29 (1971), 81–84. MR 275224, DOI 10.1090/S0002-9939-1971-0275224-9
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
G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679–698. MR 132403, DOI 10.2140/pjm.1961.11.679
D. Masson and W. K. McClary, On the self-adjointness of the $(g(x)\phi ^{4})_{2}$ Hamiltonian, Comm. Math. Phys. 21 (1971), 71–74. MR 280114, DOI 10.1007/BF01646485
Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. MR 107176, DOI 10.2307/1970331
A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6 (1965), 179–191 (1965). MR 194899, DOI 10.1007/BF02591357
A. E. Nussbaum, A note on quasi-analytic vectors, Studia Math. 33 (1969), 305–309. MR 251554, DOI 10.4064/sm-33-3-305-309
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193–254. MR 104919, DOI 10.1090/S0002-9947-1959-0104919-1
Barry Simon, The theory of semi-analytic vectors: A new proof of a theorem of Masson and McClary, Indiana Univ. Math. J. 20 (1970/71), 1145–1151. MR 290173, DOI 10.1512/iumj.1971.20.20105