Coburn's Lemma and the Finite Section Method for Random Jacobi Operators

Abstract

We study the spectra and pseudospectra of finite and infinite tridiagonal random matrices, in the case where each of the diagonals varies over a separate compact set, say $U,V,W\subset\mathbb{C}$. Such matrices are sometimes termed stochastic Toeplitz matrices $A_+$ in the semi-infinite case and stochastic Laurent matrices $A$ in the bi-infinite case. Their spectra, $\Sigma=$ spec $A$ and $\Sigma_+=$ spec $A_+$, are independent of $A$ and $A_+$ as long as $A$ and $A_+$ are pseudoergodic (in the sense of E.B. Davies, Commun. Math. Phys., 2001), which holds almost surely in the random case. This was shown in Davies (2001) for $A$; that the same holds for $A_+$ is one main result of this paper. We give upper and lower bounds on $\Sigma$ and $\Sigma_+$, and we explicitly compute a set $G$ that fills the gap between the two in the sense that $\Sigma\cup G=\Sigma_+$. We show that invertibility of one operator $A_+$ implies invertibility - and uniform boundedness of the inverses - of all finite square matrices with three diagonals in $U, V$ and $W$. This implies that the so-called finite section method for the approximate solution of a system $A_+x=b$ is applicable as soon as $A_+$ is invertible, and that the same method for estimating the spectrum of $A_+$ does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. One of our main tools is a new version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over $U$, $V$ and $W$. This allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.