A1 Refereed original research article in a scientific journal
On infinite matrices, Schur products and operator measures
Authors: Kiukas J, Lahti P, Pellonpaa JP
Publisher: PERGAMON-ELSEVIER SCIENCE LTD
Publication year: 2006
Journal: Reports on Mathematical Physics
Journal name in source: REPORTS ON MATHEMATICAL PHYSICS
Journal acronym: REP MATH PHYS
Volume: 58
Issue: 3
First page : 375
Last page: 393
Number of pages: 19
ISSN: 0034-4877
DOI: https://doi.org/10.1016/S0034-4877(06)80959-6
Abstract
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.
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.
Keywords:
covariant operator measure, extensible operator measure, generalized operator measure, generalized vector, norm of a Schur multiplier, quantum observable, Schur multiplier, Schur product
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