Condition numbers of large Toeplitz-like matrices (Q2784658)

For an invertible operator \(A\) in \(\ell_2\) the number \(\kappa(A)=\|A\|\|A^{-1}\|\) is referred to as the (spectral) condition number of \(A\). NEWLINENEWLINENEWLINELet \(T_n(a)\) be the \(n\times n \) Toeplitz matrix generated by a rational function \(a\) without poles on the unit circle. The authors study the behavior of the condition numbers NEWLINE\[NEWLINE\kappa(T_n(a)+ F_n)NEWLINE\]NEWLINE as \(n\to\infty\) for some class of perturbations \(F_n\). Namely, \(F_n\) may, in particular, include \(n\times n\) truncations of rational Hankel matrices or, for example, the cases of constant blocks of finite dimensions in the upper left and lower right corners and zeroes elsewhere. The authors take \(F_n\) in the form \(P_nKP_n + W_nLW_n\), where \(P_n\) is the projector in \(\ell_2\) onto the first \(n\) coordinates and \(W_n\) is the operator NEWLINE\[NEWLINEW_n : (x_0,x_1,x_2, \dots) \mapsto (x_{n-1}, \dots , x_1, x_0, 0,0, \dots)NEWLINE\]NEWLINE and \(K\) and \(L\) belong to the class of matrices \(K\) satisfying the condition that there exists a \(\sigma\in(0,1)=\sigma_K\) such that NEWLINE\[NEWLINE|K_{jk}|\leq C_K \sigma^{j+k} \quad \text{for all} \quad j,k\geq 0.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINELet \(T_n(\widetilde a)\) be the transpose of \(T_n(a)\) and \(M=\max(\|T(a)+K\|,\|T(\widetilde a)+L\|)\). Under the assumption that the value of \(M^{(-1)}=\max(\|(T(a)+K)^{-1}\|,\|(T(\widetilde a)+L)^{-1}\|)\) is finite, the authors prove a theorem for the operator \(A_n=P_nKP_n + W_nLW_n\) stating that always NEWLINE\[NEWLINE|\kappa(A_n)-MM^{(-1)}|= O\left(\frac{\log n}{n}\right)NEWLINE\]NEWLINE but in general, there exist a \(\gamma>0\) such that NEWLINE\[NEWLINE|\kappa(A_n)-MM^{(-1)}|= O\left(e^{-\gamma \sqrt n}\right)NEWLINE\]NEWLINE except for some exceptional cases. A comprehensive historical background enlightening the problem is also given.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00034].