On the Stampfli point of some operators and matrices

TL;DR

This paper investigates the properties and location of the Stampfli point for various classes of operators and matrices, revealing conditions under which it lies in the convex hull of the spectrum and its relation to orthogonality.

Contribution

It provides new results on the location of the Stampfli point for specific operator classes and explores its connection to spectral properties and orthogonality.

Findings

Stampfli point for almost normal operators lies in the convex hull of the spectrum.

Location results for 2-by-2 block and 3-by-3 matrices with repeated eigenvalues.

Relations between Stampfli point being zero and Roberts orthogonality.

Abstract

The center of mass of an operator $A$ (denoted St($A$), and called in this paper as the {\em Stampfli point} of A) was introduced by Stampfli in his Pacific J. Math (1970) paper as the unique $\lambda\in\mathbb C$ delivering the minimum value of the norm of $A-\lambda I$. We derive some results concerning the location of St($A$) for several classes of operators, including 2-by-2 block operator matrices with scalar diagonal blocks and 3-by-3 matrices with repeated eigenvalues. We also show that for almost normal $A$ its Stampfli point lies in the convex hull of the spectrum, which is not the case in general. Some relations between the property St($A$)=0 and Roberts orthogonality of $A$ to the identity operator are established.