# A Hilbert space approach to fractional difference equations

## TL;DR

This paper develops a Hilbert space framework for fractional difference equations, establishing existence, stability, and causality properties using functional calculus and operator theory.

## Contribution

It introduces a novel functional analytical approach to fractional difference equations, linking fractional sums to operator powers and analyzing solution stability.

## Key findings

- Existence of solutions on Hilbert space-valued weighted sequence spaces
- A stability condition for linear fractional difference equations
- Relation of fractional sums to fractional powers of operators

## Abstract

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator $1 - \tau^{-1}$ with the right shift $\tau^{-1}$ on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.