 
Summary: DIAGONALS OF SELFADJOINT OPERATORS
WILLIAM ARVESON* AND RICHARD V. KADISON
Abstract. The eigenvalues of a selfadjoint 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 SchurHorn theorem characterizes that relation in
terms of a system of linear inequalities.
We prove an extension of the latter result for positive traceclass 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
selfadjoint operators in II1 factors to their images under a conditional
expectation onto a maximal abelian subalgebra.
1. Introduction
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
SchurHorn theorem (for matrices) to operators on an infinitedimensional
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].
