On large Toeplitz band matrices with an uncertain block
Let \(a(e^{it})= \sum^q_{k=-p} a_ke^{ikt}\) be a trigonometric polynomial on the unit circle of the plane, where \(p\), \(q\geq 1\) and \(a_{-p}\), \(a_q \neq 0\). Let \(T(a)= [a_{j-\ell}]^\infty_{j,\ell=1}\) denote the infinite band nontriangular Toeplitz matrix acting on \(\ell^2(N)\) corresponding to \(a\), and, for any \(n\geq 1\), let \(T_n(a)\) be the finite Toeplitz matrix \([a_{j-\ell} ]^n_{j, \ell=1}\) on \(\mathbb{C}^n\). The main result of this paper (Theorem 2.2) proves that, for any fixed \(m\geq 1\), there exists an \(\varepsilon_2>0\) which depends on \(a\) and \(m\) such that for any \(\varepsilon\), \(0<\varepsilon< \varepsilon_2\) the structured pseudospectrum \(sp^m_\varepsilon T_n(a)\) converges to \(\Lambda(a)\) in the Hausdorff metric as \(n\) approaches infinity, where \(sp^m_\varepsilon T_n(a)\) is defined as the union of the spectra of \(T_n(a)+P_m KP_m\) with \(K\) ranging over all operators on \(\mathbb{C}^n\) with norm less than or equal to \(\varepsilon\) and \(P_m\) being the orthogonal projection to the first \(m\) components, and \(\Lambda(a)\) denotes the Schmidt-Spitzer set defined as the intersection of the spectra of \(T(a_\rho)\) with \(a_\rho(e^{it})= \sum^q_{k=-p} a_k \rho^k e^{ikt}\) for all \(\rho>0\). This shows that the small-norm perturbation in the upper-left \(m\)-by-\(m\) block of a band nontriangular Toeplitz matrix has an asymptotically stable spectrum. This is in contrast to the case for \(T(a)\) itself, in which case there is some \(\varepsilon_1>0\) such that \(sp^m_\varepsilon T(a)\) is equal to the spectrum of \(T(a)\) for all \(\varepsilon\), \(0<\varepsilon <\varepsilon_1\).
infinite band nontriangular Toeplitz matrix
scientific article; zbMATH DE number 1915799
scientific article