Abstract:

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H, I be the identity operator on H, T∈B(H). Let N(·) be an arbitrary norm on B(H). An extension of the numerical radius based on the approximate D-orthogonality by $w_{N-D-\varepsilon}(T)=\sup \left\{|\zeta|: \zeta \in \mathbf{C}, I \perp_{D-\varepsilon}^N(T-\zeta I)\right\}$ is given. It is proved that w_N-D-ε(·) is a semi-norm on B(H). It is also given a necessary and sufficient condition that w_N-D-ε(·) is a norm on B(H). When w_N-D-ε(·) is a norm, the geometry and related properties of the normed linear space (B(H), w_N-D-ε(·)) are investigated.

Key words: N-D-ε orthogonality, N-D-ε numerical radius, norm, bounded linear operator, normed linear space