Shift operators and their classification

TL;DR

This paper introduces and classifies a broad class of shift operators on Banach spaces, providing conditions for their dynamical properties and linking hyperbolicity with the shadowing property.

Contribution

It defines a new class of shift operators, classifies many families, and establishes a connection between hyperbolicity and shadowing in their dynamics.

Findings

Classified various families of shift operators.

Provided verifiable conditions for dynamical properties.

Linked generalized hyperbolicity with the shadowing property.

Abstract

We introduce a class of linear bounded invertible operators on Banach spaces, called shift operators, which comprises weighted backward shifts and models finite products of weighted backward shifts and dissipative composition operators. We classify vast families of these shift operators, including the ones generated by orthogonal, diagonalizable, rotation or hyperbolic matrices. and this classification yields verifiable conditions which we use to construct concrete examples of shift operators with a variety of dynamical properties. As a consequence, we show that, for large classes of shift operators, generalized hyperbolicity is equivalent to the shadowing property.