Diagonals of operators Majorization, a Schur{Horn theorem and zero-diagonal idempotents A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematical Sciences McMicken College of Arts and Sciences University of Cincinnati, June 2016 Author: Jireh Loreaux Chair: Gary Weiss, Ph.D. Degrees: B.A. Mathematics, 2010, Committee: Jonathan Brown, Ph.D. University of Cincinnati Victor Kaftal, Ph.D. Costel Peligrad, Ph.D. Shuang Zhang, Ph.D. Abstract This dissertation investigates two related but mostly independent topics. A diagonal of an operator is the sequence which appears on the diagonal of the matrix representation with respect to a prescribed orthonormal basis. The classical Schur{Horn Theorem provides a characterization of the set of diagonals of a n n selfadjoint operator on C (or R ) in terms of its eigenvalues. The relation which provides this description is called majorization. New kinds of majorization (p-majorization and approximate p-majorization) are defined herein, and it is shown that they characterize the diagonals of positive compact operators with infinite dimensional kernel. When the kernel is finite dimensional the diagonals lie between two closely related classes of sequences: those whose zero set differs in cardinality from the zero set of the eigenvalues by p and are also p-majorized (or in the case of the second set, approximately p-majorized) by the eigenvalue sequence. These results extend the work of Kaftal and Weiss in 2010 which describes precisely the diagonals of those positive compact operators with kernel zero. Zero-diagonal operators are those which have a constant zero diagonal in some or- thonormal basis. Several equivalences are obtained herein for an idempotent operator to be zero-diagonal including one which is basis independent, answering a 2013 question of Jasper about the existence of nonzero zero-diagonal idempotents. These techniques are extended to prove that any bounded sequence of complex numbers appears as the diagonal of some idempotent, and that the diagonals of finite rank idempotent operators are those complex-valued absolutely summable sequences whose sum is a positive integer. ii c 2016 by Jireh Loreaux. All rights reserved. Acknowledgments The author was supported by the University of Cincinnati Department of Mathematical Sciences, the Charles Phelps Taft Research Center, the Choose Ohio First Scholarship Program, as well as the following conferences: Non-Commutative Geometry and Operator Algebras; Southeastern Analysis Meeting; Great Plains Operator Theory Symposium; International Conference on Operator Theory; East Coast Operator Algebras Symposium; Wabash Modern Analysis Seminar; and Singular Traces: Theory and Applications hosted by the Centre International de Rencontres Math´ematiques. I would like to thank my advisor, Gary Weiss, for his continual prodding, thorough feedback and valuable insights, without which I might never have completed this disserta- tion. I am appreciative of the candor with which he communicates both commendation and exhortation. I am also indebted to him for our many disagreements over writing style and mathematical reasoning which resulted in more economical, precise and lucid exposition. Moreover, his support is a primary reason I remained at the University of Cincinnati for graduate school. I would like to acknowledge Victor Kaftal for devoting his time and energy to instruct me on the subject of von Neumann algebras even when no formal course could be offered within the department. I must also thank Shuang Zhang for siphoning time from his busy schedule as Department Head to teach me the basics of C∗-algebras. iv I owe my gratitude to my committee for bothering to read, edit and critique this dissertation. I thank the constituents of the functional analysis seminar for listening to me ramble seemingly unceasingly concerning these results. I am grateful to Daniel Beltit¸˘afor introducing me to the work of Fan, Fong and Herrero, without which my research on diagonals of idempotents may never have flourished. I am indebted to John Jasper for suggesting to me at GPOTS 2013 several of the major problems of study in this dissertation. I thank Jean-Christophe Bourin for communicating that Theorem 4.2.7 is a corollary of his pinching theorem [Bou03, Theorem 2.1]. My friends in the department, especially Sasmita Patnaik, Marcos Lopez, Michael Snook, Dewey Estep, Seth Ball and Catalin Dragan deserve special mention for listening to my complaints and providing ample distraction from work when the need arose. Their laughter and companionship were essential to slogging through coursework and research daily. I owe Michael Snook a great deal for the many fruitful discussions concerning libre software, GNU/Linux, cryptography, and of course, LATEX. I would not be where I am today were it not for my parents, Richard and Debra Loreaux, who instilled in me a passion for mathematics and made great sacrifices to provide me with the best opportunities to pursue this career. Their love, support and encouragement have been mainstays during my journey long before I entered the University of Cincinnati. And finally, the most influential people in my life: my wife, Katie, and my daughters Dara and Anastacia. Words cannot adequately express Kate's role in this arena. She pushed me onward when I felt as though I wanted to quit; she never doubted me for a second and encouraged me daily; most importantly, she has never given up on me in spite of my many failings. As for my daughters, they have inspired me and made sure I always had my hands full. Their smiles and laughter provide me with an unending source of joy and happiness. \Great are the works of the Lord; They are studied by all who delight in them." 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