Classes of integrable metrics on Banach spaces

Abstract

We introduce a large new class of (weak) metrics on the Hardy space that have infinite families of conservation laws in involution. The construction of the metrics and the integrals involve the Hankel operator and a field of Nijehuis operators. The commutativity of the integrals is based on the notion of the coupling transform, which allows us to recursively construct new classes of metrics and integrals. We characterize the spectrum of the Nijenhuis operators. The paper establishes a connection between a rich class of completely integrable geodesic flows appearing in the finite dimensional Riemannian geometry and a new class of integrable systems in Banach spaces.