A class of tridiagonal operators associated to some subshifts

Abstract

: We consider a class of tridiagonal operators induced by not necessary pseudoergodic biinfinite sequences. Using only elementary techniques we prove that the numerical range of such operators is contained in the convex hull of the union of the numerical ranges of the operators corresponding to the constant biinfinite sequences; whilst the other inclusion is shown to hold when the constant sequences belong to the subshift generated by the given biinfinite sequence. Applying recent results by S. N. Chandler-Wilde et al. and R. Hagger, which rely on limit operator techniques, we are able to provide more general results although the closure of the numerical range needs to be taken.

The numerical range of a class of periodic tridiagonal operators

“…From [11, Theorem 2.5], we already know that the numerical range of T is contained in the set Γ equal to the convex hull of the union of the numerical ranges of the tridiagonal operators T (0, 0, 1) and T (1, 0, 1). In fact, by [11,Corollary 2.7], this set Γ is the numerical range of some tridiagonal operator. When a is the 2-periodic sequence of period 01, we prove in this section a similar result: the closure of the numerical range of T is the closure of the convex hull of the union of the numerical ranges of two matrices in M 2 (C).…”

“…where the rectangle marks the matrix entry at (0, 0). When A is the set {−1, 1}, the corresponding operator A b is related to the so called "hopping sign model" introduced in [7] and subsequently studied in many other works, such as [1,[3][4][5][9][10][11], just to name a few.…”

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

“…Consider A to be a finite set of complex numbers and let a = (a i ) i∈Z be a biinfinite sequence in the total shift space A Z . In [12], the tridiagonal operator A a : ℓ 2 (Z) → ℓ 2 (Z) associated to a is defined as ( 1)…”

“…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.…”

The numerical range of some periodic tridiagonal operators is the convex hull of the numerical ranges of two finite matrices

“…They also have found applications in Physics. For example, a tridiagonal operator is used as the "hopping sign model" introduced in [8] and studied by many other authors, such as in [2][3][4][5][9][10][11]. Though it may seem natural to try finding the numerical range of arbitrary tridiagonal operators and matrices, this turns out to be a hard problem.…”