Slant joint antieigenvalues and antieigenvectors of operators in normal subalgebras

We will study the slant joint antieigenvalues and antieigenvectors of pairs of operators that belong to the same closed normal subalgebra of the algebra of bounded operators on a separable Hilbert space. This extends the slant antieigenvalue theory from single normal operators to pairs of normal operators. Our results may be viewed as extensions of the Greub-Rheinboldt inequality from two positive operators to two normal operators.     

Antieigenvalue techniques in statistics

In a number of his recent papers Karl Gustafson has outlined the similarities between the Antieigenvalue Theory he founded and several finite dimensional matrix optimization theorems for positive matrices arising in statistics. In this paper, we will show how the techniques that the author and Karl Gustafson have used for computation of Antieigenvalues can also be applied to prove and generalize these matrix optimization theorems in statistics. We will primarily focus on two techniques which we have used in Antieigenvalue computations in recent years. These two techniques are a two nonzero component property for certain class of functionals, and converting the matrix optimization problems in statistics to a convex programing problem. Indeed, these two techniques allow us to generalize some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces.