Schur-Horn Theorem Research Articles

Geometric proofs of some theorems of Schur-Horn type

We describe elementary, geometric proofs of the classical Schur-Horn theorem for symmetric matrices and analogous results for skew-symmetric matrices, in the real and complex cases. We compare this approach to the proofs obtained by making use of symplectic techniques. As a further application, we present a new proof of Horn's characterization of diagonal entries of orthogonal matrices.

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.

Changes in dermal vascularity

Schur–Horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes, formally described as the range of the canonical conditional expectation of its unitary orbit.Following a brief background survey, we prove an infinite dimensional Schur–Horn theorem for positive compact operators with infinite dimensional kernel, one of the two open cases posed recently by Kaftal–Weiss. There, they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators. Here we show how the characterization problem depends on the dimension of the kernel when it is finite or infinite dimensional.We obtain exact majorization characterizations of the range of the canonical conditional expectation of the unitary orbits of positive compact operators with infinite dimensional kernel, unlike the approximate characterizations of Arveson–Kadison, but extending the exact characterizations of Gohberg–Markus and Kaftal–Weiss.Recent advances in this subject and related subjects like traces on ideals show the relevance of new kinds of sequence majorization as in the work of Kaftal–Weiss (e.g., strong majorization and another majorization similar to what here we call p-majorization), and of Kalton–Sukochev (e.g., uniform Hardy–Littlewood majorization), and of Bownik–Jasper (e.g., Riemann and Lebesgue majorization). Likewise key tools here are new kinds of majorization, which we call p- and approximate p-majorization (0≤p≤∞).