Antieigenvalue techniques in statistics
Given an operator \(T\) on a Hilbert space \(H\), the first antieigenvalue of \(T\), also called the (real) cosine of \(T\), denoted by \(\mu_1(T)\) or \(\cos T\), was first defined by \textit{K. Gustafson} [Bull. Am. Math. Soc. 74, 488--492 (1968; Zbl 0172.40702)] to be \[ \mu_1(T)= \inf_{Tf\neq 0}\frac{\text{Re}(Tf,f)}{\| Tf\|\, \| f\|}\,. \] The author continues to explore two techniques used in antieigenvalue computations in previous papers. The first is basically a two nonzero component property for certain class of functionals. The second consists on converting the matrix optimization problems in statistics to a convex programming problem, finding the minimum of a convex function on the numerical range of an operator. The two techniques will permit generalizing some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces.
scientific article; zbMATH DE number 5552958