Open Access
April 2017 Radon–Nikodym theorems for operator-valued measures and continuous generalized frames
Fengjie Li, Pengtong Li
Banach J. Math. Anal. 11(2): 363-381 (April 2017). DOI: 10.1215/17358787-0000008X


In this article we determine that an operator-valued measure (OVM) for Banach spaces is actually a weak∗ measure, and then we show that an OVM can be represented as an operator-valued function if and only if it has σ-finite variation. By the means of direct integrals of Hilbert spaces, we introduce and investigate continuous generalized frames (continuous operator-valued frames, or simply CG frames) for general Hilbert spaces. It is shown that there exists an intrinsic connection between CG frames and positive OVMs. As a byproduct, we show that a Riesz-type CG frame does not exist unless the associated measure space is purely atomic. Also, a dilation theorem for dual pairs of CG frames is given.


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Fengjie Li. Pengtong Li. "Radon–Nikodym theorems for operator-valued measures and continuous generalized frames." Banach J. Math. Anal. 11 (2) 363 - 381, April 2017.


Received: 1 March 2016; Accepted: 9 June 2016; Published: April 2017
First available in Project Euclid: 22 February 2017

zbMATH: 1383.43003
MathSciNet: MR3612170
Digital Object Identifier: 10.1215/17358787-0000008X

Primary: 43A15
Secondary: 42C15 , 46B22 , 46G10 , 47A20

Keywords: continuous generalized frame , dilation , operator-valued measure , Radon–Nikodym theorem , weak∗ measure

Rights: Copyright © 2017 Tusi Mathematical Research Group

Vol.11 • No. 2 • April 2017
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