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 wN-D-ε(T)=sup{|ζ|:ζ∈C,I⊥ND-ε(T-ζI)}is given.It is proved that wN-D-ε(·)is a semi-norm on B(H).It is also given a neces-sary and sufficient condition that wN-D-ε(·)is a norm on B(H).When wN-D-ε(·)is a norm,the geometry and related properties of the normed linear space(B(H),wN-D-ε(·))are investigated.
N-D-ε orthogonalityN-D-ε numerical radiusnormbounded linear operatornormed linear space
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