On the structure of shift-invariant subspaces of \(L^ 2 (T^ 2, \mu)\)
Suppose that \(V_1\) and \(V_2\) are commuting isometries on a Hilbert space \({\mathcal H}\). In [\textit{G. Kallianpur} and \textit{V. Mandrekar}, Prediction theory and harmonic analysis, Pesi Masani Vol., 165-190 (1983; Zbl 0522.60050)] and [\textit{M. Slocinski}, Ann. Polon. Math. 37, 255-262 (1980; Zbl 0485.47018)] conditions are given for \({\mathcal H}\) to have four-fold Wold and Halmos decompositions with respect to \(V_1\) and \(V_2\). These notions are used by \textit{V. Mandrekar} [Proc. Am. Math. Soc. 103, No. 1, 145-148 (1988; Zbl 0658.47033)] and \textit{J. Radlow} [Proc. Am. Math. Soc. 38, 293-297 (1973; Zbl 0266.47018)] to characterize invariant subspaces \({\mathcal M}\) of the Hardy space \(H^2(T^2)\) which are generated by an inner function. Such \({\mathcal M}\) are shown to be those invariant subspaces on which \(V_1\) and \(V_2\) doubly commute, where \(V_j\) is now multiplication by the coordinate variable \(z_j\), \(j=1,2\). More generally, \textit{P.Ghatage} and \textit{V. Mandrekar} [J. Operator Theory 20, No. 1, 83-89 (1988; Zbl 0703.46038)] describe the invariant subspaces of \(L^2(T^2)\) on which these \(V_1\) and \(V_2\) doubly commute. In this article we explore these ideas in the case \({\mathcal M}\) is an invariant subspace of the weighted space \(L^2(T^2,\mu)\).
four-fold Wold and Halmos decompositions
scientific article; zbMATH DE number 757774