On~a~class of extremal problems

A. L. Sakhnovich

Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 411-418

Résumé

The matrix-valued function $\rho(\lambda,\mu)=\Phi_2(E-\lambda A^*)^{-1}S^{-1}(E-\mu A)^{-1}\Phi_2$ is investigated for operators $S>0$ satisfying the operator identity $AS-SA^*=i\Pi_1/\Pi_1^*$, $\Pi_1=[\Phi_1,\Phi_2]$. Connected with the operator $S$ is the problem of describing the taxation functions (nondecreasing operator-valued functions) $\sigma$ giving the representations $S=\int_{-\infty}^\infty(E-At)^{-1}\Phi_2\,d\sigma(t)\Phi_2^*(E-A^*t)^{-1}$. It is proved that the maximal jump in taxation functions at a point $\lambda_0$ ($\operatorname{Im}{\lambda_0}=0$) is equal to $\rho^{-1}(\lambda_0,\lambda_0)$. The asymptotic behavior of $\rho_k(\lambda_0,\overline\lambda_0)$ for $\operatorname{Im}{\lambda_0}\geqslant0$ as $k\to\infty$ is studied in the case when a sequence of operators $S_k$ acting in spaces $H_k$ ($H_1\subset H_2\subset\cdots$) is given. In the case of Toeplitz matrices $S$ the asymptotic behavior of $\rho_k(\lambda_0, \overline\lambda_0)$ yields the first limit theorem of Szegë. Bibliography: 19 titles.