On infinite matrices, Schur products and operator measures

: Kiukas J, Lahti P, Pellonpaa JP

Publisher: PERGAMON-ELSEVIER SCIENCE LTD

: 2006

Reports on Mathematical Physics

REPORTS ON MATHEMATICAL PHYSICS

: REP MATH PHYS

: 58

: 3

: 375

: 393

: 19

: 0034-4877

DOI: https://doi.org/10.1016/S0034-4877(06)80959-6

Measures with values in the set of sesquilinear forms on a subspace of a Hilbert space are of interest in quantum mechanics, since they can be interpreted as observables with only a restricted set of possible measurement preparations. In this paper, we consider the question under which conditions such a measure extends to an operator-valued measure, in the concrete setting where the measure is defined on the Borel sets of the interval [0, 2 pi) and is covariant with respect to shifts. In this case, the measure is characterized with a single infinite matrix, and it turns out that a basic sufficient condition for the extensibility is that the matrix be a Schur multiplier. Accordingly, we also study the connection between the extensibility problem and the theory of Schur multipliers. In particular, we define some new norms for Schur multipliers.