On the computation of entropy functionals and their minima in indeterminate extension problems. (Russian) Zbl 0556.47013

Arov, D. Z.; Kreĭn, M. G.

Acta Sci. Math. 45, 33-50 (1983).

Let \(M_{m,n}\) denote the complex \(m\times n\)-matrices. For each measurable \(M_{m,n}\)-valued function f on \(\{\zeta \in {\mathbb{C}};| \zeta | =1\}\) with \(\| f(\zeta)\| \leq 1\) almost everywhere and for each \(z\in {\mathbb{C}}\) with \(| z| <1\) one sets \[ i(f;z)=- \frac{1}{4\pi}\int_{| \zeta | =1}\ln \det [I_ n- f(\zeta)^*f(\zeta)]\frac{1-| z|^ 2}{| \zeta -z|^ 2}| d\zeta |. \] Let further \(A=(a_{ik})_{i,k=1,2}\) be a measurable \(\left( \begin{matrix} I_ m\\ 0\end{matrix} \begin{matrix} 0\\ -I_ n\end{matrix} \right)\)-unitary valued function on \(\{\zeta \in {\mathbb{C}};| \zeta | =1\}\), such that \(\chi =-a^{- 1}_{22}a_{21}\) has a bounded analtic extension on \(\{\zeta \in {\mathbb{C}};| \zeta | <1\}\) and \(i(\chi;0)<+\infty.\) Let us denote \(f_ A=(a_{11}f+a_{12})(a_{21}f+a_{22})^{-1}.\) It is proved, that for all analytic \(M_{m,n}\)-valued f on \(\{\zeta \in {\mathbb{C}};| \zeta | <1\}\) with \(\| f\|_{\infty}\leq 1\) holds \[ i(f_ A;z)=i(\chi;z)+i(f;z)+\ln | \det (I_ n-\chi (z)f(z))|. \] Moreover, \(i(f_ A;z)\) is minimized only by \(f\equiv \chi (z)^*.\)
These results are applied to the computation of the minimum of \(i(f;z)\) for all measurable \(M_{m,n}\)-valued functions f on \(\{\zeta \in {\mathbb{C}};| \zeta | =1\}\), for which \(\| f(\zeta)\| \leq 1\) almost everywhere and whose principal part \(\sum^{\infty}_{k=1}\gamma_ k\zeta j^{-k}\) is given. Analogous results are proved also for functions on the upper half plane.

Reviewer: L.Zsido

MSC:

47A56	Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A65	Structure theory of linear operators
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A20	Dilations, extensions, compressions of linear operators