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Some Banach spaces of measurable operator-valued functions

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Bibliographic Details
Title: Some Banach spaces of measurable operator-valued functions
Authors: Klotz, Lutz
Publisher Information: University of Wrocław (Uniwersitet Wrocławski), Wrocław
Subject Terms: Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.), Stochastic processes, Spaces of vector- and operator-valued functions, nonnegative matrix-valued measures, square integrable operator-valued functions, infinite-dimensional stationary processes, multivariate stationary processes, Convergence of probability measures, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), stationary stochastic processes
Description: The space \(L^ 2(\mu)\) of square integrable scalar-valued functions with respect to a nonnegative real-valued measure has played an important role in the analysis of univariate 2nd order stationary stochastic processes. This space was generalized by several authors in late 1950s and early 1960s to consist of square integrable matrix-valued functions with respect to nonnegative matrix-valued measures; its properties such as completeness and the denseness of simple functions were used to study the time domain analysis of multivariate stationary processes [\textit{M. Rosenberg}, Duke Math. J. 31, 291-298 (1964; Zbl 0129.08902)]. In the 1970s this space was further generalized and was defined to consist of square integrable operator-valued functions with respect to a nonnegative operator-valued measure; again its fundamental properties: the denseness of simple functions and its completeness property were used to investigate the structure of the time domain of infinite-dimensional stationary processes [\textit{V. Mandrekar} and the reviewer, Indiana Univ. Math. J. 20, 545-563 (1970; Zbl 0252.46040)]. In the present article the author studies Banach spaces of \(p\)-integrable operator-valued function with respect to an operator-valued measure. These spaces are natural generalization of the standard spaces \(L^ p(\mu)\) of \(p\)-integrable scalar-valued functions with respect to nonnegative real-valued measure; and for the case \(p=2\) they reduce to the space of square integrable operator-valued functions mentioned above.
Document Type: Article
File Description: application/xml
Access URL: https://zbmath.org/167564
Accession Number: edsair.c2b0b933574d..b6a98d2d7cb69552ec12e9bc80b15b1b
Database: OpenAIRE
Description
Abstract:The space \(L^ 2(\mu)\) of square integrable scalar-valued functions with respect to a nonnegative real-valued measure has played an important role in the analysis of univariate 2nd order stationary stochastic processes. This space was generalized by several authors in late 1950s and early 1960s to consist of square integrable matrix-valued functions with respect to nonnegative matrix-valued measures; its properties such as completeness and the denseness of simple functions were used to study the time domain analysis of multivariate stationary processes [\textit{M. Rosenberg}, Duke Math. J. 31, 291-298 (1964; Zbl 0129.08902)]. In the 1970s this space was further generalized and was defined to consist of square integrable operator-valued functions with respect to a nonnegative operator-valued measure; again its fundamental properties: the denseness of simple functions and its completeness property were used to investigate the structure of the time domain of infinite-dimensional stationary processes [\textit{V. Mandrekar} and the reviewer, Indiana Univ. Math. J. 20, 545-563 (1970; Zbl 0252.46040)]. In the present article the author studies Banach spaces of \(p\)-integrable operator-valued function with respect to an operator-valued measure. These spaces are natural generalization of the standard spaces \(L^ p(\mu)\) of \(p\)-integrable scalar-valued functions with respect to nonnegative real-valued measure; and for the case \(p=2\) they reduce to the space of square integrable operator-valued functions mentioned above.