Positive forms and dilations
Author:
Wacław Szymański
Journal:
Trans. Amer. Math. Soc. 301 (1987), 761780
MSC:
Primary 47A20; Secondary 42A70, 43A35, 47D05
MathSciNet review:
882714
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References 
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Additional Information
Abstract: By using the quadratic form and unbounded operator theory a new approach to the general dilation theory is presented. The boundedness condition is explained in terms of the Friedrichs extension of symmetric operators. Unbounded dilations are introduced and discussed. Applications are given to various problems involving positive definite functions.
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 [1]
 T. Ando and W. Szymanski, Order structure and Legesgue decomposition of positive definite operator functions, Indiana Univ. Math. J. 35 (1986), 157173. MR 825633 (87k:47050)
 [2]
 W. B. Arveson, Subalgebras of algebras, Acta Math. 123 (1969), 142224. MR 0253059 (40:6274)
 [3]
 J. Glimm and A. Jaffe, Quantum physics, SpringerVerlag, New York, 1981. MR 628000 (83c:81001)
 [4]
 P. R. Halmos, A Hilbert space problem book, Graduate Texts in Math. 19, SpringerVerlag, New York, 1980. MR 675952 (84e:47001)
 [5]
 , How not to solve the invariant subspace problem, a lecture during the ``Asymmetric Algebras and Invariant Subspaces'' conference, Bloomington, Indiana, March, 1986.
 [6]
 W. Mlak, Dilations of Hilbert space operators (general theory), Dissertationes Math. 153 (1978), 165. MR 496046 (81j:47008)
 [7]
 W. Mlak and A. Weron, Dilations of Banach space operator valued functions, Ann. Polon. Math. 38 (1980), 295303. MR 599254 (82b:47003)
 [8]
 M. Reed and B. Simon, Methods of modern mathematical physics. I, II, Academic Press, New York, 1972, 1975. MR 751959 (85e:46002)
 [9]
 Z. Sebestyen, Moment theorems for operators on Hilbert space, Acta Sci. Math. (Szeged) 44 (1982), 165171. MR 660523 (84f:47004)
 [10]
 , Moment theorems for operators on Hilbert space. II, Acta Sci. Math. (Szeged) 47 (1984), 101106. MR 755567 (86b:47065)
 [11]
 F. H. Szafraniec, Dilations on involution semigroups, Proc. Amer. Math. Soc. 66 (1977), 3032. MR 0473873 (57:13532)
 [12]
 B. Sz.Nagy, Extensions of linear transformations in Hilbert spaces which extend beyond the space, Appendix to F. Riesz and B. Sz.Nagy, Functional Analysis, Ungar, New York, 1960.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708827140
PII:
S 00029947(1987)08827140
Keywords:
Positive quadratic forms,
the Friedrichs extension of a positive operator,
positive definite function,
dilation,
dilation,
moment problem,
reconstruction of quantum mechanics
Article copyright:
© Copyright 1987
American Mathematical Society
