Unitary orbits of Hermitian operators with convex or concave functions

Convex and concave relaxations of implicit functions

On convex-concave perception based functions

Perception based function (PBF) is given by the set of rules Ri:“If X is Ti then Y is Si”, where Ti is a linguistic term describing some fuzzy intervals Ai on the domain of real values of X and Si is a linguistic description of the shape of the function Y(X) on this interval. The methods of explicitation of such type of rules when Si are given by words QUICKLY INCREASING AND SLIGHTLY CONVEX, SL...

Unitary Orbits of Normal Operators in Von Neumann Algebras

The starting points for this paper are simple descriptions of the norm and strong* closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neu...

The Sum of Unitary Similarity Orbits Containing Only Special Operators

Dedicated to Professor Shmuel Friedland. Abstract Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A1, . . . , Ak ∈ B(H) such that for any unitary operators U1, . . . , Uk, ∑k j=1 U ∗ j AjUj is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. ...

On Self-Concordant Convex-Concave Functions

In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of “good enough” convex-concave functions – those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an ...