On the relation between positive definite functions and generalized Toeplitz kernels

Friedrich, J.

doi:10.1090/S0002-9939-1994-1209422-X

On the relation between positive definite functions and generalized Toeplitz kernels
HTML articles powered by AMS MathViewer

by J. Friedrich PDF

Proc. Amer. Math. Soc. 120 (1994), 727-730 Request permission

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 operator-valued measures.

References

Rodrigo Arocena, On the extension problem for a class of translation invariant positive forms, J. Operator Theory 21 (1989), no. 2, 323–347. MR 1023319
R. Arocena and M. Cotlar, Dilation of generalized Toeplitz kernels and some vectorial moment and weighted problems, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 169–188. MR 654185
Rodrigo Arocena and Mischa Cotlar, Generalized Toeplitz kernels and Adamjan-Arov-Kreĭn moment problems, Toeplitz centennial (Tel Aviv, 1981) Operator Theory: Advances and Applications, vol. 4, Birkhäuser, Basel-Boston, Mass., 1982, pp. 37–55. MR 669900
Ramón Bruzual, Local semigroups of contractions and some applications to Fourier representation theorems, Integral Equations Operator Theory 10 (1987), no. 6, 780–801. MR 911991, DOI 10.1007/BF01196119
Mischa Cotlar and Cora Sadosky, On the Helson-Szegő theorem and a related class of modified Toeplitz kernels, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 383–407. MR 545279
Jürgen Friedrich, Integral representations of positive definite matrix-valued distributions on cylinders, Trans. Amer. Math. Soc. 313 (1989), no. 1, 275–299. MR 992599, DOI 10.1090/S0002-9947-1989-0992599-1

On representations of positive definite operator functions

17