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.
Highlights
About the scalar complex trigonometric moment problem we recall that: a sequence tn n Z of complex numbers with tn t n is called positive semi-definite if for each n 0, the Toeplitz matrixTn ti j n i, j 0 is positive semi-definite
[10], another proof of a quite similar necessary and sufficient existence condition on a sequence of bounded operators to admit an integral representation as trigonometric moment sequence with respect to some positive operator valued measure is given
We extend Ai to Ki1 preserving the above definition and boundedness condition; the extensions Ai : Ki1 Ki2,1 i p are denoted with the same letter Ai In case that
Summary
About the scalar complex trigonometric moment problem we recall that: a sequence tn n Z of complex numbers with tn t n is called positive semi-definite if for each n 0 , the Toeplitz matrix. In [4], the necessary and sufficient condition of representing a finite sequence of bounded operators on an arbitrary Hilbert space H, m Aokm ekn n tn with An A n , A0 IdH sequence is the positivity of as a trigonometric the Toeplitz matrix. In [9], Corollary 1.4.10., a necessary and sufficient condition for solving a trigonometric operator-valued moment problem is given. In [10], another proof of a quite similar necessary and sufficient existence condition on a sequence of bounded operators to admit an integral representation as trigonometric moment sequence with respect to some positive operator valued measure is given. In Proposition 3.2, Section 3, the necessary and sufficient existence condition in Proposition 3.1 is stated in terms of matrices
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