The Schur-Horn Theorem in infinite dimensions and related problems
by
Victor Kaftal
University of Cincinnati

A sequence x is called admissible for a positive bounded operator A if in the SOT, A=∑x_i P_i for some sequence of rank-one projections. Equivalently, if x is the diagonal of WAW^* for some partial isometry with W^*W the range projection of A.

If A is compact and Tr(A)=∞ then x is admissible for A if and only if x is not summable and ∑_j=1ⁿ x^*_n ≤ ∑_j=1ⁿ s(A)_j for every n, where x^* and s(A) are the monotone rearrangements of x and of the sequence of eigenvalues of A. If Tr(A) < ∞ we have to further ask that ∑_j=1^∞ x_n = Tr(A). This Schur-Horn theorem is joint work with Gary Weiss.

An "inverse" problem is finding the operators for which < 1, 1, ... > is admissible, i.e., that are strong sums of projections. This is obtained for operators in von Neumann factors and in the multiplier algebra of a purely infinite simple s-unital C* algebras (joint work with Ping Wong Ng and Shuang Zhang)

A necessary and sufficient condition for a sequence to be admissible for a projection was obtained by R. Kadison, The Pythagorean Theorem I & II, Proc. Natl. Acad. Sci. USA 99 (7) (2002), 4178-4184, 8 (2002), 5217-5222. The same condition is also sufficient for strong sum of projections (joint work with David Larson.)

Date received: June 13, 2008