Diagonals of Operators: Majorization, a Schur-Horn Theorem and Zero-Diagonal Idempotents

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 selfadjoint operator on ℂⁿ (or ℝⁿ) 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 orthonormal 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.