Majorization and a Schur–Horn theorem for positive compact operators, the nonzero kernel case

Abstract

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 KaftalWeiss. 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 ≤ ∞).

In search of convexity: diagonals and numerical ranges

“…Despite the convexity of D(T ) may, in general, fail even for self-adjoint T , it was proved that compact positive operators, which are either of finite rank or of infinite rank and with infinitedimensional kernel, have convex sets of diagonals. See [24,Corollary 6.7] and [31,Corollary 4.3] for these results.…”

“…While Kadison's theorem concerns the set {D(P ) : P is a self-adjoint projection on H}, it is easy to adopt it to our framework of fixed P (as observed in [32, p. 94] Despite the convexity of D(T ) may, in general, fail even for self-adjoint T , it was proved that compact positive operators, which are either of finite rank or of infinite rank and with infinitedimensional kernel, have convex sets of diagonals. See [24,Corollary 6.7] and [31,Corollary 4.3] for these results.…”

On restricted diagonalization

“…This, in turn, allowed the characterization of those finite spectrum normal operators that are U J (H)-diagonalizable provided by Loreaux in the same work [22]. It is interesting to notice that the motivations for this characterization are related with Arveson's index obstruction [5,25] satisfied by the diagonals of certain normal operators with finite spectrum, which is well known to be a generalization of Kadison's integer condition [24] for the diagonals of self-adjoint projections (for related work on diagonals of operators and index obstructions see [6,12,26,29,30]). Moreover, following the line of previous research on these topics, Loreaux showed the deep relations between the study of several questions raised in [10] and the work [13] on unitary equivalence modulo compact operators.…”

Submajorization on $\ell^p(I)^+$ determined by increasable doubly substochastic operators and its linear preservers

“…In recent years, there is a big progress towards developing various extensions of the most important majorization relations on sequence spaces [5,15] and on descrete Lebesgue spaces [6,7,8,9,10,18,19,20,21,22,23] with apropriate generalizations of some famous theorems in linear algebra [2,3,4,16,24,27,30,32]. There are a lot of applications of the majorization theory in various branches of mathematics and there exist significant conections with the other science like physics, quantum mechanics and quantum information theory [12,17,25,30,31,33].…”