Summary: DIAGONALS OF SELF-ADJOINT OPERATORS
WILLIAM ARVESON* AND RICHARD V. KADISON
Abstract. The eigenvalues of a self-adjoint n×n matrix A can be put
into a decreasing sequence = (1, . . . , n), with repetitions according
to multiplicity, and the diagonal of A is a point of Rn
that bears some
relation to . The Schur-Horn theorem characterizes that relation in
terms of a system of linear inequalities.
We prove an extension of the latter result for positive trace-class op-
erators on infinite dimensional Hilbert spaces, generalizing results of one
of us on the diagonals of projections. We also establish an appropriate
counterpart of the Schur inequalities that relate spectral properties of
self-adjoint operators in II1 factors to their images under a conditional
expectation onto a maximal abelian subalgebra.
This paper presents some of the results of a project begun by the authors
that is directed toward finding an appropriate common generalization of the
Schur-Horn theorem (for matrices) to operators on an infinite-dimensional
Hilbert space, and to operators in finite factors, in a form that would gen-
eralize work of one of us on projections in II1 factors [Kad02a], [Kad02b].