dorsal/arxiv
View SchemaOn infinite matrices, Schur products, and operator measures
| Authors | J. Kiukas, P. Lahti, J. -P. Pellonpää |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0609060 |
| URL | https://arxiv.org/abs/quant-ph/0609060 |
| DOI | 10.1016/S0034-4877(06)80959-6 |
| Journal | Rep. Math. Phys. 58 (2006) 375-393 |
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.
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"abstract": "Measures with values in the set of sesquilinear forms on a subspace of a\nHilbert space are of interest in quantum mechanics, since they can be\ninterpreted as observables with only a restricted set of possible measurement\npreparations. In this paper, we consider the question under which conditions\nsuch a measure extends to an operator valued measure, in the concrete setting\nwhere the measure is defined on the Borel sets of the interval $[0,2\\pi)$ and\nis covariant with respect to shifts. In this case, the measure is characterized\nwith a single infinite matrix, and it turns out that a basic sufficient\ncondition for the extensibility is that the matrix be a Schur multiplier.\nAccordingly, we also study the connection between the extensibility problem and\nthe theory of Schur multipliers. In particular, we define some new norms for\nSchur multipliers.",
"arxiv_id": "quant-ph/0609060",
"authors": [
"J. Kiukas",
"P. Lahti",
"J. -P. Pellonp\u00e4\u00e4"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/S0034-4877(06)80959-6",
"journal_ref": "Rep. Math. Phys. 58 (2006) 375-393",
"title": "On infinite matrices, Schur products, and operator measures",
"url": "https://arxiv.org/abs/quant-ph/0609060"
},
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