A class of tridiagonal operators associated to some subshifts

TL;DR

This paper investigates the numerical range of a class of tridiagonal operators linked to subshifts, establishing bounds and relations using elementary and advanced limit operator techniques.

Contribution

It introduces new bounds for the numerical range of these operators and connects elementary methods with recent limit operator results for broader applicability.

Findings

Numerical range is contained in the convex hull of constant sequence operators' ranges.

Equality holds when constant sequences are in the subshift generated by the sequence.

Results extend previous work using elementary and limit operator techniques.

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.