Applications of autoreproducing kernel Grammian moduli to \({\mathcal S}(U,H)\)-valued stationary random functions
Let \((X_ g)_{g\in G}\) be a stationary process over a LCA group G with values in the space of all bounded linear operators from a Hilbert space H into a Hilbert space K. The time domain M(X) of \((X_ g)_{g\in G}\) is the least subspace of K generated by all \(X_ gx\), \(g\in G\), \(x\in H\). If \((X_ g)_{g\in G}\) is additionally a continuous function from G into the set of Hilbert-Schmidt operators, HS(H,K), equipped with the Hilbert-Schmidt norm, then instead of M(X), the time domain \({\mathcal M}(X)\) is customarily defined to be a closed Hilbertian submodulus of HS(H,K) generated by the operators of the form \(X_ gA\), \(g\in G\), A is linear and bounded in H. The isomorphic description of \({\mathcal M}(X)\) and M(X) were provided by \textit{V. Mandrekar} and the reviewer [Indiana Univ. Math. J. 20, 545-563 (1970; Zbl 0252.46040), J. Multivariate Anal. 1, 167-185 (1971; Zbl 0218.46040)] and by \textit{A. Makagon} [ibid. 14, 114-133 (1984; Zbl 0536.60052)], respectively. The author of the reviewed paper studies Hilbert-Schmidt operator valued stationary processes. Based on the notion of an autoreproducing property the author produces a Hilbertian modulus which may serve as a model for the time domain M(X) of a stationary process. A brief analysis of the relationships between this construction and the works mentioned above is included. In the remaining part of the paper the decomposition of the modulus constructed in view of the Lebesgue-Cramér decomposition of a process and some topics related to the interpolation problem are discussed.
Hilbert-Schmidt operator valued stationary processes
Lebesgue-Cramér decomposition of a process
scientific article; zbMATH DE number 3949402
scientific article