A-NUMERICAL RADIUS OF A-NORMAL OPERATORS IN SEMI-HILBERTIAN SPACES

Abstract

Let \(\mathcal{H}\) be a Hilbert space and A be a positive bounded linear operator on \(\mathcal{H}\).  The semi-inner product ⟨h, k⟩_A = ⟨Ah, k⟩, h, k ∈ \(\mathcal{H}\), induces a seminorm for a bounded linear operator T, which is defined by

                   ∥T∥_A = sup{∥Th∥_A/∥h∥_A : ∥h∥ ≠ 0}.

The main purpose of this paper is to prove that ∥T∥_A equals the A-numerical radius of T , when T is an A-normal operator.  This generalizes the similar result for a normal operator on a Hilbert space.