Positive-Definite Operator-Valued Kernels and Integral Representations

L. Lemnete-Ninulescu

Departament of Mathematics, Politechnica University of Bucharest, Bucharest, Romania.

DOI: 10.4236/am.2012.312274                

Abstract

A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let be a finite sequence of bounded operators, with arbitrary, acting on a finite dimensional Hilbert space H. A necessary and sufficient condition on the positivity of an operator kernel for the existence of an atomic, positive, operator-valued measure , with the property that for every with , the moment of coincides with the term of the sequence, is given. The connection between some positive definite operator-valued kernels and the Riesz-Herglotz integral representation of the analytic on the unit disc, operator-valued functions with positive real part in the class of operators in Section 4 of the note is studied.

Keywords

Unitary-Operator; Self-Adjoint Operator; Joint Spectral Measure of a Commuting Tuple of Operators; Spectral Projector; Complex Moments; Analytic Vectorial Functions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. Carathéodory, “über den Variabilitatsbereich der Fourierschen Konstanten von Positiven Harmonischen Funktionen,” Rendiconti del Circolo Matematico di Palermo, Vol. 32, No. 1, 1911, pp. 193-207. doi:10.1007/BF03014795
[2]	G. Herglotz, “über Potenzreihen mit Positivem, Reelem Teil im Einheitskreis,” Leipziger Berichte, Mathematics, Physics, Vol. 63, 1911, pp. 501-511.
[3]	C. Carathéodory und L. Fejér, “über den Zusammenhang der Extreme von Harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz,” Rendiconti del Circolo Matematico di Palermo, Vol. 32, No. 1, 1911, pp. 218-239. doi:10.1007/BF03014796
[4]	T. Ando, “Truncated Moment Problems for Operators,” Acta Mathematica, Vol. 31, 1970, pp. 319-334.
[5]	L. Lemnete-Ninulescu, “Truncated Trigonometric and Hausdorff Moment Problems for Operators,” Proceedings of the 23th International Operator Conference, Timisoara, 29 June-4 July 2010, pp. 51-61.
[6]	M. Bakonyi and V. Lopushanskaya, “Moment Problems for Real Measures on the Unit Circle,” Operator Theory Advances and Applications, Vol. 198, 2009, pp.49-60.
[7]	F. J. Narcowich, “R-Operators II., on the Approximation of Certain Operator-Valued Analytic Functions and the Hermitian Moment Problem,” Indiana University Mathematics Journal, Vol. 26, No. 3, 1977, pp. 483-513. doi:10.1512/iumj.1977.26.26038
[8]	M. Putinar and F. H. Vasilescu, “Solving Moment Problems by Dimensional Extension,” Annals of Mathematics, Vol. 148, No. 3, 1999, pp. 1087-1107.
[9]	F. H. Vasilescu, “Spectral Measures and Moment Problems,” In: Spectral Theory and Its Applications, Theta, Bucharest, 2003, pp. 173-215.
[10]	L. Lemnete-Ninulescu, “Positive-Definite Operator-Valued Functions and the Moment Problem,” Operator Theory Live, Proceedings of the 22th International Operator Conference, Timisoara, 3-8 July 2008, 2012, pp. 113-123.
[11]	J. W. Helton and M. Putinar, “Positive Polynomials in Scalar and Matrix Variables, the Spectral Theorem, and Optimization,” In: Operator Theory, Structured Matrices, and Dilations, Theta Series in Advanced Mathematics, Theta, Bucharest, 2007, pp. 229-307.
[12]	N. I. Akhiezer, “The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburgh, 1965.