TY - JOUR

T1 - On Infinite Matrices, Schur Products and Operator Measures

AU - Kiukas, Jukka

AU - Lahti, Pekka

AU - Pellonp, Juha Pekka

N1 - Funding Information:
The authors thank Prof. Karl Ylinen for fruitful discussions. One of us (J.K.) was supported by Turku University Foundation.
Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2006/12/1

Y1 - 2006/12/1

N2 - 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 defned on the Borel sets of the interval [0, 2π) 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.

AB - 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 defned on the Borel sets of the interval [0, 2π) 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.

KW - covariant operator measure

KW - extensible operator measure

KW - generalized operator measure

KW - generalized vector

KW - norm of a Schur multiplier

KW - quantum observable

KW - Schur multiplier

KW - Schur product

UR - http://www.scopus.com/inward/record.url?scp=34548227901&partnerID=8YFLogxK

U2 - 10.1016/S0034-4877(06)80959-6

DO - 10.1016/S0034-4877(06)80959-6

M3 - Article

AN - SCOPUS:34548227901

SN - 0034-4877

VL - 58

SP - 375

EP - 393

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

IS - 3

ER -