On the relation between positive definite functions and generalized Toeplitz kernels
Author:
J. Friedrich
Journal:
Proc. Amer. Math. Soc. 120 (1994), 727730
MSC:
Primary 42A82; Secondary 47A57, 47B35
MathSciNet review:
1209422
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Abstract: We show that extension problems for generalized Toeplitz kernels may be completely reduced to extension problems for positive definite functions, where the solution is well known. These considerations in particular imply that generalized Toeplitz kernels may be represented as Fourier transforms of positive operatorvalued measures.
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 [1]
 R. Arocena, On the extension problem for a class of translation invariant positive forms, J. Operator Theory 21 (1989), 323347. MR 1023319 (90k:47015)
 [2]
 R. Arocena and M. Cotlar, Dilation of generalized Toeplitz kernels and some vectorial moment and weighted problems, Lecture Notes in Math., vol. 908, Springer, Berlin, 1982. MR 654185 (84j:47006)
 [3]
 , Generalized Toeplitz kernels and AdamjanArovKrein moment problems, Oper. Theory: Adv. Appl., vol. 4, Birkhauser, Basel and Boston, MA, 1982, pp. 3755. MR 669900 (84a:42021)
 [4]
 R. Bruzual, Local semigroups of contractions and some applications to Fourier representation theorems, Integral Equations Operator Theory 10 (1987), 780801. MR 911991 (89a:47065)
 [5]
 M. Cotlar and C. Sadosky, On the HelsonSzegö theorem and a related class of modified Toeplitz kernels, Amer. Math. Soc., Providence, RI, 1979, pp. 383407. MR 545279 (81j:42022)
 [6]
 J. Friedrich, Integral representations of positive definite matrixvalued distributions on cylinders, Trans. Amer. Math. Soc. 313 (1989), 275299. MR 992599 (90k:43013)
 [7]
 M. L. Gorbachuk, On representations of positive definite operator functions, Ukrain. Mat. Z. 17 (1965), 2945. (Russian)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919941209422X
PII:
S 00029939(1994)1209422X
Article copyright:
© Copyright 1994
American Mathematical Society
