The numerical range of a class of periodic tridiagonal operators

Abstract

In this paper we compute the closure of the numerical range of certain periodic tridiagonal operators. This is achieved by showing that the closure of the numerical range of such operators can be expressed as the closure of the convex hull of the uncountable union of numerical ranges of certain symbol matrices. For a special case, this result can be improved so that it is the convex hull of the union of the numerical ranges of only two matrices. A conjecture is stated for the general case.

The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix

“…If n > 1, for each φ ∈ [0, 2π), following [1,13] we define the symbol of T , as the following (n + 1) × (n + 1) matrix ( 2)…”

“…In the particular case of the alphabet A = {−1, 1}, the corresponding operator A a is related to the so called "hopping sign model" introduced in [7] and subsequently studied in many other works, such as [1,2,3,4,5,6,8,9,12], just to name a few. On the other hand, when the alphabet is A = {0, 1} some results for computing the numerical range of A a are presented in [12,13]. In particular, work in [13] addresses the case when a is an n + 1-periodic sequence.…”

“…On the other hand, when the alphabet is A = {0, 1} some results for computing the numerical range of A a are presented in [12,13]. In particular, work in [13] addresses the case when a is an n + 1-periodic sequence. Relying on the fact that the closure of the numerical range of A a may be written as the closure of the convex hull of an uncountable union of numerical ranges of certain matrices, in [13] the closure of the numerical range of the 2-periodic case is computed by substituting such uncountable union of numerical ranges by the convex hull of the union of the numerical ranges of just two 2×2 matrices.…”

“…In particular, work in [13] addresses the case when a is an n + 1-periodic sequence. Relying on the fact that the closure of the numerical range of A a may be written as the closure of the convex hull of an uncountable union of numerical ranges of certain matrices, in [13] the closure of the numerical range of the 2-periodic case is computed by substituting such uncountable union of numerical ranges by the convex hull of the union of the numerical ranges of just two 2×2 matrices. In this work, we further contribute to the study of the numerical range of A a when a is an n + 1 periodic biinfinite sequence.…”

“…In this work, we further contribute to the study of the numerical range of A a when a is an n + 1 periodic biinfinite sequence. Instead of working with the operators A a , we work with the more general tridiagonal operators T = T (a, b, c) defined in Section 2, since, as can be seen in [13], the computation of the closure of the numerical range of A a is a particular case of that of T . Using a result of Plaumann and Vinzant [18], we show that the closure of the numerical range of the Date: January 2021.…”