Convergent multiples of convergent operators Original Research Article

Let “Xmuch greater-than0” mean that “the bounded linear Hilbert-space operator X is selfadjoint, positive, and invertible”. We discuss the operators A which are known to be convergent (i.e. have spectral radius less than 1) because they all satisfy Steinʹs condition P−A*PAmuch greater-than0 for a fixed Pmuch greater-than0. We use the convexity of this set of Aʹs to show that when certain operators are in it (and hence convergent), others (often multiples) must be also. Our results generalize, and are modivated by, some results in [A. Bhaya, E. Kaszkurewicz, Linear Algebra Appl. 187 (1993) 87–104).