Gradient flows for bounded linear evolution equations

TL;DR

This paper characterizes which bounded linear evolution equations in Hilbert spaces can be expressed as gradient flows, specifically those with real diagonalisable operators, and explores related convexity properties.

Contribution

It provides a constructive characterization of bounded linear evolution equations that are gradient flows, focusing on the diagonalisability of the operator.

Findings

Equations with real diagonalisable operators can be written as gradient flows.

Constructive proof method for identifying gradient flow structures.

Derivation of geodesic lambda-convexity for these equations.

Abstract

We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambda-convexity.