A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus
\textit{B. Kostant} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 6, 413-455 (1973; Zbl 0293.22019)] proved a far-reaching generalization of the Schur-Horn theorem on eigenvalues of Hermitian matrices to the case of a general compact Lie group. The authors prove an infinite-dimensional analogue of this result, using the group \(S \text{Diff}({\mathcal A})\), where \({\mathcal A} := \{0 \leq z \leq 1\} \times \{\exp(2\pi i\theta) \mid 0 \leq \theta < 1\}\), of area-preserving diffeomorphisms of the annulus \(\mathcal A\) as an analogue of the group \(\text{SU}(n)\) in the Schur-Horn theory. This group acts on its Lie algebra, the algebra of divergence-free (Hamiltonian) vector fields tangent to the boundary of \(\mathcal A\). The Schur-Horn-Kostant theorem relies on projection onto an orbit of the Weyl group. But since the Weyl group of \(S \text{Diff}({\mathcal A})\) only has two elements, the authors are forced to pass to a completion of \(S \text{Diff}({\mathcal A})\) obtained by extending the adjoint representation to unitary operators on \(L_ 2({\mathcal A})\) and using the strong operator topology. The result is that \(S \text{Diff}\) is dense in the group \(S \text{Meas}({\mathcal A})\) of invertible measure-preserving (MP) transformations of \(\mathcal A\). The main Schur-Horn-Kostant-type result is that the projection of an adjoint orbit onto a ``Cartan'' subalgebra isomorphic to \(L_ 2([0,1])\) is an infinite-dimensional, weakly compact, convex set, whose extreme points coincide with the orbit (through a certain function) of the ``permutation'' semigroup of MP transformations of [0,1]. The denseness result follows from a result of S. Alpern that an invertible MP transformation of the square can be approximated by a MP homeomorphism that fixes the boundary of the square (and hence of the annulus) and a result of J. Moser showing that such a homeomorphism of the square can be approximated by a measure-preserving diffeomorphism. The Schur-Horn result relies on results of Ryff on doubly stochastic operators.
approximation of measure-preserving transformations
divergence-free (Hamiltonian) vector fields
scientific article; zbMATH DE number 549664
scientific article