## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 11, Number 2 (2017), 363-381.

### Radon–Nikodym theorems for operator-valued measures and continuous generalized frames

#### Abstract

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 $\sigma $-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.

#### Article information

**Source**

Banach J. Math. Anal., Volume 11, Number 2 (2017), 363-381.

**Dates**

Received: 1 March 2016

Accepted: 9 June 2016

First available in Project Euclid: 22 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1487732419

**Digital Object Identifier**

doi:10.1215/17358787-0000008X

**Mathematical Reviews number (MathSciNet)**

MR3612170

**Zentralblatt MATH identifier**

1383.43003

**Subjects**

Primary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

Secondary: 42C15: General harmonic expansions, frames 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10] 47A20: Dilations, extensions, compressions

**Keywords**

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

#### Citation

Li, Fengjie; Li, Pengtong. Radon–Nikodym theorems for operator-valued measures and continuous generalized frames. Banach J. Math. Anal. 11 (2017), no. 2, 363--381. doi:10.1215/17358787-0000008X. https://projecteuclid.org/euclid.bjma/1487732419