A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
On infinite matrices, Schur products and operator measures
Tekijät: Kiukas J, Lahti P, Pellonpaa JP
Kustantaja: PERGAMON-ELSEVIER SCIENCE LTD
Julkaisuvuosi: 2006
Lehti:Reports on Mathematical Physics
Tietokannassa oleva lehden nimiREPORTS ON MATHEMATICAL PHYSICS
Lehden akronyymi: REP MATH PHYS
Vuosikerta: 58
Numero: 3
Aloitussivu: 375
Lopetussivu: 393
Sivujen määrä: 19
ISSN: 0034-4877
DOI: https://doi.org/10.1016/S0034-4877(06)80959-6
Tiivistelmä
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.
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