On Infinite Matrices, Schur Products and Operator Measures

Jukka Kiukas*, Pekka Lahti, Juha Pekka Pellonp

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)375-393
Number of pages19
JournalReports on Mathematical Physics
Issue number3
Publication statusPublished - 01 Dec 2006
Externally publishedYes


  • 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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