Essential numerical ranges of operators in semi-Hilbertian spaces

Abstract

Let A be a positive bounded operator on an infinite dimensional complex and separable Hilbert space (H, ⟨, ⟩) and ∶=B(H)∕K(H) be the Calkin algebra and ′ its dual; here K(H) denotes the closed ideal of B(H) consisting of all compact operators on H . For an operator T on H , let ‖T‖ A and V A (T) denote the A-operator semi-norm and the A-numerical range induced by A. The A-essential numerical range and the A-essential norm of T are defined by V e A (T) = V Â( T) andwhereIn this paper, we establish some properties of the A-essential numerical range.In particular we show that if T has an A-adjoint then