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 -