# The Selgrade decomposition for linear semiflows on Banach spaces

## TL;DR

This paper extends classical dynamical systems concepts like Selgrade's Theorem and Morse spectrum to linear semiflows on Banach spaces, providing a characterization of exponential separation in infinite-dimensional settings.

## Contribution

It generalizes finite-dimensional dynamical systems results to the Banach space setting, introducing new tools for analyzing linear skew product semiflows.

## Key findings

- Characterization of exponentially separated subbundles as attractor-repeller pairs
- Extension of Selgrade's Theorem to Banach bundles
- Recovery of finite-dimensional properties in infinite-dimensional context

## Abstract

We extend Selgrade's Theorem, Morse spectrum, and related concepts to the setting of linear skew product semiflows on a separable Banach bundle. We recover a characterization, well-known in the finite-dimensional setting, of exponentially separated subbundles as attractor-repeller pairs for the associated semiflow on the projective bundle.