Approximate ω-orthogonality and ω-derivation

. We introduce the notion of approximate ω -orthogonality (referring to the numerical radius ω ) and investigate its signi fi cant properties. Let T , S ∈ B ( H ) and ε ∈ [ 0 , 1 ) . We say that T is approximate ω -orthogonality to S and we write T ⊥ εω S if ω 2 ( T + λ S ) (cid:2) ω 2 ( T ) − 2 εω ( T ) ω ( λ S ) , for all λ ∈ C . We show that T ⊥ εω S if and only if inf θ ∈ [ 0 , 2 π ) D θω ( T , S ) (cid:2) − εω ( T ) ω ( S ) in which D θω ( T , S ) = lim r → 0 + ω 2 ( T + re i θ S ) − ω 2 ( T ) 2 r ; and this occurs if and only if for every θ ∈ [ 0 , 2 π ) , there exists a sequence { x θ n } of unit vectors in H such that and where ω ( T ) is the numerical radius of T . ,