Anderson's theorem and A-spectral radius bounds for semi-Hilbertian space operators

Abstract

Let H be a complex Hilbert space and let A be a positive bounded linear operator on H. Let T be an A-bounded operator on H. For rank(A)=n<∞, we show that if WA(T)⊆D‾(={λ∈C:|λ|≤1}) and WA(T) intersects ∂D(={λ∈C:|λ|=1}) at more than n points, then WA(T)=D‾. In particular, when A is the identity operator on Cn, then this leads to Anderson's theorem in the complex Hilbert space Cn. We introduce the notion of A-compact operators to study analogous result when the space H is infinite dimensional. Further, we develop an upper bound for the A-spectral radius of n×n operator matrices with entries are commuting A-bounded operators, where A=diag(A,A,…,A) is an n×n diagonal operator matrix. Several inequalities involving A-spectral radius of A-bounded operators are also given.