Phone: 618-650-5070
Email: jloreau@siue.edu


Jireh Loreaux

Click here for a copy of my curriculum vitae. Click on paper titles for the most recent versions prior to official publication and DOIs for links to the published articles; arXiv links are also included for preprints.

Published

Victor Kaftal and Jireh Loreaux, Kadison's Pythagorean Theorem and essential codimension, Integral Equations and Operator Theory, vol. 87 (4), pp. 565–580, doi: 10.1007/s00020-017-2365-y, arXiv: 1609.06754 [math.OA], Apr. 2017.

Abstract. Kadison's Pythagorean theorem (2002) provides a characterization of the diagonals of projections with a subtle integrality condition. Arveson (2007), Kaftal, Ng, Zhang (2009), and Argerami (2015) all provide different proofs of that integrality condition. In this paper we interpret the integrality condition in terms of the essential codimension of a pair of projections introduced by Brown, Douglas and Fillmore (1973), or, equivalently of the index of a Fredholm pair of projections introduced by Avron, Seiler, and Simon (1994). The same techniques explain the integer occurring in the characterization of diagonals of selfadjoint operators with finite spectrum by Bownik and Jasper (2015).



Jireh Loreaux and Gary Weiss, Diagonality and idempotents with applications to problems in operator theory and frame theory, Journal of Operator Theory, vol. 75 (1), pp. 91–118, doi: 10.7900/jot.2014nov05.2054, arXiv: 1410.7441 [math.FA], Feb. 2016.

Abstract. We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert–Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis.

We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.



Jireh Loreaux and Gary Weiss, Majorization and a Schur–Horn Theorem for positive compact operators, the nonzero kernel case, Journal of Functional Analysis, vol. 268 (3), pp. 703–731, doi: 10.1016/j.jfa.2014.10.020, Feb. 2015.

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 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 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\le p\le \infty $).

Accepted

Jireh Loreaux and Gary Weiss, Traces on ideals and the commutator property, Dec. 2017, Proceedings of the 26th International Conference on Operator Theory (to appear).

Abstract. We propose a new class of traces motivated by a trace/trace class property discovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products of operators outside the trace class. Spectral traces, traces that depend only on the spectrum and algebraic multiplicities, possess this property and we suspect others do, but we know of no other traces that do.

This paper is intended to be part survey. We provide here a brief overview of some facts concerning traces on ideals, especially involving Lidskii formulas and spectral traces.

We pose the central question: whenever the relevant products, $AB,BA$ lie in an ideal, do bounded operators $A,B$ always commute under any trace on that ideal, i.e., $\tau (AB) = \tau (BA)$? And if not, characterize which traces/ideals do possess this property.



John Jasper, Jireh Loreaux and Gary Weiss, Thompson's theorem for compact operators and diagonals of unitary operators, Indiana University Mathematics Journal (to appear), arXiv: 1602.08435 [math.FA], Apr. 2016.

Abstract. As applications of Kadison's Pythagorean and carpenter's theorems, the Schur–Horn theorem, and Thompson's theorem, we obtain an extension of Thompson's theorem to compact operators and use these ideas to give a characterization of diagonals of unitary operators. Thompson's mysterious inequality concerning the last terms of the diagonal and singular value sequences plays a central role.

Submitted

Jireh Loreaux, Restricted diagonalization of finite spectrum normal operators and a theorem of Arveson, arXiv: 1712.06554 [math.FA], Dec. 2017, submitted to Journal of Operator Theory.

Abstract. Kadison characterized the diagonals of projections and observed the presence of an integer, which Arveson later recognized as a Fredholm index obstruction applicable to any normal operator with finite spectrum coincident with its essential spectrum whose elements are the vertices of a convex polygon. Recently, in joint work with Kaftal, the author linked the Kadison integer to essential codimension of projections.

This paper provides an analogous link between Arveson's obstruction and essential codimension as well as an entirely new proof of Arveson's theorem which also allows for generalization to any finite spectrum normal operator. In fact, we prove that Arveson's theorem is a corollary of a trace invariance property of arbitrary normal operators. An essential ingredient is a nontrivial operator-theoretic formulation of Arveson's theorem in terms of restricted diagonalization, that is, diagonalization by unitaries which are Hilbert--Schmidt perturbations of the identity.

Department of Mathematics and Statistics
Southern Illinois University Edwardsville
Vadalabene Center

35 Circle Drive
Edwardsville, Illinois 62026

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