An operator-valued generalization of Tchakaloffʼs theorem

Let S:={St1,…,td}0⩽t1+⋯+td⩽mS:={St1,…,td}0⩽t1+⋯+td⩽m be a finite multisequence of p×pp×p Hermitian matrices. If there exists a p×pp×p positive semidefinite matrix-valued Borel measure Σ   on RdRd so thatSt1,…,td=∫⋯∫Rdx1t1⋯xdtddΣ(x1,…,xd), for all d  -tuples of non-negative integers (t1,…,td)(t1,…,td) so that 0⩽t1+⋯+td⩽m0⩽t1+⋯+td⩽m, i.e. Σ is a representing measure S  , then we will show that there exist p×pp×p positive semidefinite matrices P1,…,PkP1,…,Pk and x1,…,xk∈suppΣ so that ∑q=1kδxqPq is also a representing measure for S  , where ∑q=1krankPq⩽p2(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 Tchakaloffʼs theorem holds. We will make use of an operator-valued generalization of Tchakaloffʼs theorem on the unit circle to obtain a solution to the operator-valued Carathéodory interpolation problem.