Poisson geometry of the infinite Toda lattice

Many important conservative systems have a non canonical Hamiltonian formulation in terms of Lie-Poisson brackets. For integrable systems, this is usually the first of two or more compatible brackets. With few notable exceptions, such as the Euler, Poisson-Vlasov, KdV, or sine-Gordon equations, for example, for infinite dimensional systems this Lie-Poisson bracket formulation is mostly formal. It is our belief that these formal approaches can be given a solid functional analytic underpinning. The present paper formulates such an approach for the infinite Toda lattice. It raises fundamental issues about the nature of coadjoint orbits for certain Banach Lie groups and it poses questions about the nature of the integrability of the Toda lattice in infinite dimensions by providing a functional analytic framework in which these problems can be rigorously formulated. The background of the present work is the paper by Odzijewicz and Ratiu [2003] where the theory of Banach Lie-Poisson spaces was developed. The paper is organized as follows. Section 2 recalls the minimal necessary material from the aforementioned paper necessary for this work. Section 3 introduces the Banach Lie group of upper bidiagonal bounded operators and discusses the Lie-Poisson structure on the predual of its Lie algebra, the Banach space of lower bidiagonal trace class operators. Generic coadjoint orbits of this group are studied in detail in Section 3. The