Nonstationary analogues of the Herglotz representation theorem for unbounded operators
It is well known that every Carathéodory function \(\phi(z)\), i.e., a function which is analytic and has a positive real part in the open unit disk, admits a Herglotz integral representation \[ \phi(z)=\text{Im }\phi(0)+\int_0^{2\pi}(1+ze^{-it})(1-ze^{-it})^{-1} d\sigma(t)\tag{*} \] for some nonnegative finite measure \(d\sigma\). Formula (*) can be rewritten in the form \[ \phi(z)=\text{Im }\phi(0)+C(I+zU)(I-zU)^{-1}C^*,\tag{**} \] where a linear functional \(C:L_2(d\sigma)\longrightarrow {\mathbb C}\) is defined by \(Cf=\int_0^{2\pi}f(t) d\sigma(t)\), and \(U\) is the operator of multiplication by \(e^{-it}\) in \(L_2(d\sigma)\), which is obviously unitary. Passing to Taylor's coefficients \(\phi_k,\;k=0,1,\ldots,\) of \(\phi(z)\), one can derive the expression analogous to the right-hand side of formula (**) for the bounded upper triangular Toeplitz operator \(\Phi=[\phi_{i-j}]_{i,j\in {\mathbb Z}}\) on the space \(l^2({\mathbb Z})\) with a positive definite real part; here \(\phi_k=0\) for \(k<0\). A generalization of this analogue of formula (**) to bounded upper triangular, not necessarily Toeplitz, operators with positive semidefinite real parts, which is important for nonstationary linear systems theory, was given by \textit{D.~Alpay, A.~Dijksma} and \textit{Y.~Peretz} [J. Funct. Anal. 166, 85-129 (1999; Zbl 0984.47008)]. In the paper under review, the authors extend the notion of upper triangular operator to unbounded operators, define the notion of Carathéodory operator, which is a (not necessarily bounded) upper triangular operator with positive semidefinite real part, and obtain an analogue of formula (**) for Carathéodory operators.
scientific article; zbMATH DE number 1798874