A Blaschke-type condition and its application to complex Jacobi matrices

Abstract

We obtain a Blaschke-type necessary condition on zeros of analytic functions on the unit disc with different types of exponential growth at the boundary. These conditions are used to prove Lieb-Thirring-type inequalities for the eigenvalues of complex Jacobi matrices.

Some Remarks on Upper Bounds for Weierstrass Primary Factors and Their Application in Spectral Theory

“…where R p is an explicitly known constant and a n (K) denotes the nth approximation number of K. For other appearances of Γ p in eigenvalue estimates (sometimes with a different notation), see, e.g. [4,8,12,25,22,9,11,16,32,13,24,14,15,17,23]. The results from below will allow us to compute the Γ p 's numerically (apparently, this has not been done before).…”

Perturbation determinants in Banach spaces — with an application to eigenvalue estimates for perturbed operators

“…We conclude this section by noting that there is another (slightly different) way to obtain estimates on the discrete spectrum of . For instance, if is a real interval and we could prove a bound of the form then we could invoke a theorem of Borichev, Golinskii and Kupin (which deals with the distribution of zeros of holomorphic functions on the unit disk which grow exponentially on the unit circle, with different rates of growth for different points of the circle), to obtain more precise bounds on the number of eigenvalues of in . Of course, to obtain a bound of the above form we would need much more information on the operators involved.…”

“…As an example, let us take a look at the well-studied case of operators acting on a complex Hilbert space H (see [14,Chapter IV.2] or [32]). Here, assuming that the operator L is an element of the nth von Neumann-Schatten class S n (H), n ∈ N, (which consists of those compact operators on H whose singular values are in l n (N)), one defines the n-regularized determinant 1…”

Lieb–Thirring inequalities for complex finite gap Jacobi matrices

“…In recent years, several results have also been established for non-selfadjoint perturbations of selfadjoint Jacobi matrices [1,[16][17][18][19]. For compact non-selfadjoint perturbations J = J 0 + δ J of the free Jacobi matrix J 0 , a near generalization (with an extra ε) of the Lieb-Thirring bound (1.2) was obtained by Hansmann and Katriel [19] using the complex analytic approach developed in [1]. Their non-selfadjoint version of the Lieb-Thirring inequalities takes the following form: For every 0 < ε < 1,…”

“…Our goal is to obtain Lieb-Thirring inequalities for complex perturbations of periodic and, more generally, almost periodic Jacobi operators with absolutely continuous finite gap spectrum. Lieb-Thirring inequalities for selfadjoint and complex perturbations of the discrete Laplacian have been studied extensively in the last decade [1,7,15,17,19,21]. The original work of Lieb and Thirring [25,26] was carried out in the context of continuous Schrödinger operators, motivated by their study of the stability of matter.…”