An operator-valued generalization of Tchakaloff's theorem

摘要

Let S:= {S_(t_1),...,t_d}0≤(t_1)+...(t_d)≤m be a finite multisequence of p x p Hermitian matrices. If there exists a, p x p positive semidefinite matrix-valued Borel measure ∑ on R~d so that for all d-tuples of non-negative integers (t_1,..., t_d) so that 0 ≤ t_1 +... + t_d ≤ m,i.e. ∑ is a representing measure S, then we will show that there exist p x p positive semidefinite matrices P_1,..., P_k and x_1.,x_k ε suppzy so that ∑_(q=1)~2 is also a representing measure for S, where ∑_(q=1)~k rank P_q ≤ p~2 (m + d)!/(m!d!). We will pose a necessary and sufficient condition on a given sequence S, of bounded linear operators on a separable Hilbert space, so that an operator-valued generalization of Tchakaloff's theorem holds. We will make use of an operator-valued generalization of Tchakaloff's theorem on the unit circle to obtain a solution to the operator-valued Carathéodory interpolation problem.