On the Stampfli point of some operators and matrices

Abstract

The center of mass of an operator A (denoted St(A) , and called in this paper as the Stampfli point of A ) was introduced by Stampfli in his Pacific J. Math (1970) paper as the unique λ ∈ C delivering the minimum value of A − λ 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.