Discrete and continuous frames can be considered as positive operator-valued measures (POVMs)
that have integral representations using rank-one operators. However, not every
POVM has an integral representation. One goal of this paper is to examine
the POVMs that have finite-rank integral representations.
More precisely, we present a necessary and sufficient condition under
which a positive operator-valued measure
$F: \Omega \to B(H)$ has an integral representation of the form
$$F(E) =\sum_{k=1}^{m} \int_{E}\, G_{k}(\omega)\otimes G_{k}(\omega) d\mu(\omega)$$
for some weakly measurable maps $G_{k} \ (1\leq k\leq m) $ from a measurable space
$\Omega$ to a Hilbert space
$\mathcal{H}$ and some positive measure
$\mu$ on $\Omega$. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM
can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.
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On Parseval Frames of Kernel Functions in de Branges Spaces of Entire Vector Valued Functions
We consider the existence and structure properties of Parseval frames of kernel functions in vector valued de Branges spaces. We develop some sufficient conditions for Parseval sequences by identifying the main construction with Naimark dilation of frames. The dilation occurs by embedding the de Branges space of vector valued functions into a dilated de Branges space of vector valued functions. The embedding also maps the kernel functions associated with a frame sequence of the original space into a Riesz basis for the embedding space. We also develop some sufficient conditions for a dilated de Branges space to have the Kramer sampling property.
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- NSF-PAR ID:
- 10314053
- Date Published:
- Journal Name:
- New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative
- Volume:
- 286
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-76473-9_1
