**Diagonals of positive operators, an alternative approach.**

I will survey the problem of finding the diagonals diag ξ of a positive operator A, from the classic Schur-Horn theorem to the more recent Kadison Pythagorean theorem.

Recognizing that this problem is equivalent to finding the coefficients in the decomposition of A= ∑_{j}ξ_{j}P_{j} where P_{j} are rank-one projections, provides an additional, sometimes easier, approach.

An illustration of this fact is the new proof of the sufficiency condition in the Kadison Pythagorean theorem, not only for the case that A is a projection, but also when A is a sum of (not necessarily mutually orthogonal) projections.

The talk is based on joint work with David Larson.

- R. Kadison: The Pythagorean Theorem II: the infinite discrete case,
*Proc. Natl. Acad. Sci. USA*, 99 (2002), 5217-5222. - V. Kaftal & J. Loreau: Kadison’s Pythagorean Theorem and essential codimension,
*Int Eq. Oper. Th*, 87 (2017), 565-580. - V. Kaftal & D. Larson: Admissible sequences of positive operators. To appear
*Trans AMS*.