  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.
