Abstract
We prove that every measurable positive definite generalized Toeplitz Kernel, defined in an (finite or infinite) interval (-a,a), is the sum of a positive definite generalized Toeplitz kernel given by continuous functions and a positive definite generalized Toeplitz kernel which vanishes almost everywhere. The proof is based on the theory of local semigroups of contractions developed in former works. In the case of ordinary Topeplitz kernels this result gives theorems of F. Riesz, M. Krein and M. Crum and a special case of a theorem of Z. Sasvári.
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References
Arocena, R. and Cotlar, M. Continuous generalized Toeplitz kernels inR. Portugaliae Math. 39, 419–434 (1980)
Arocena, R. and Cotlar, M. Generalized Toplitz kernels and Adamjam-Arov-Krein moment problems. Op. theory: Advances and applications 4, 37–55 (1982).
Aronszan, N. Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950).
Bruzual, R. Local semigroups of contractions and some applications to Fourier representation theorems. Int. Eq. and Op. Theory, Vol. 10, 780–801 (1987).
Bruzual, R. Unitary extensions of two parameter local semigroups of isometric operators and the Krein extension theorem. Int. Eq. and Op. Theory, Vol. 17, 301–322 (1993).
Bruzual, R. and Marcantognini, S. Local semigroups of isometries in Πκ -spaces and related continuation problems for κ-indefinite Toeplitz kernels. Int. Eq. and Op. Theory, Vol. 15, 527–550 (1992).
Cotlar, M. and Sadosky, C.: On the Helson-Szegö theorem and a related class of modified Toeplitz kernels. Proc. Symp. Pure Math. AMS, 35.1, 383–407 (1979).
Cotlar, M. and Sadosky, C. Prolongements des formes de Hankel generalisess en formes de Toeplitz. C. R. Acad. Sci. Paris, t. 305, Serie I, 167–170 (1987)
Cotlar, M. and Sadosky, C. Two-parameter lifting theorems and double Hilbert transforms in commutative and non commutative settings. J. Mathematical Analysis and Applications 150, 439–480 (1990).
Crum, M.: On positive definite functions. Proc. London Math. Soc. (3) 6, 548–560 (1956).
Dunford, N. and Schwartz, J. Linear Operators. Part I. Interscience 1957
Grossmann, M. and Langer, H.: Uber indexerhaltende Erweiterungen eines hermiteschen operators in Pontrjaginraum. Math. Nachrichten 64, 289–317 (1974).
Krein, M. G.: On the representation of functions by Fourier Stieltjes integrals. (Russian). Učenije Zapiski Kuibishevskogo Gosud. Pedag. i Učitelskogo Inst. 7, 123–148 (1943).
Krein, M. G.: On measurable hermitian positive functions. (Russian) Mat. Zametki 23, 79–89 (1978). English translation: Math. notes 23, 45–50 (1978)
Sasvári, Z.: Positive definite and definitizable functions. Akademie Verlag (1994)
Riesz, F.: Über Sätze von Stone und Bochner. Acta Sci. Math. 6, 184–198 (1932–1934).
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Bruzual, R. Representation of measurable positive definite generalized Toeplitz Kernels inR . Integr equ oper theory 29, 251–260 (1997). https://doi.org/10.1007/BF01320699
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DOI: https://doi.org/10.1007/BF01320699