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Numerical index with respect to an operator

The concept of numerical index was introduced by Lumer in 1968 in the context of the study and classification of operator algebras. This is a constant relating the norm and the numerical range of bounded linear operators on the space. More precisely, the numerical index of a Banach space X, n⁢(X), is the greatest constant k≥0 such that k∥T∥≤sup{|x∗(Tx)|:x∗∈X∗,x∈X,∥x∗∥=∥x∥=x∗(x)=1} for every T∈ℒ⁢(X).

Recently, Ardalani introduced new concepts of numerical range, numerical radius, and numerical index, which generalize in a natural way the classical ones and allow to extend the setting to the context of operators between possibly different Banach spaces. Given a norm-one operator G∈ℒ⁢(X,Y) between two Banach spaces X and Y, the numerical index with respect to G, nG⁢(X,Y), is the greatest constant k≥0 such that k∥T∥≤infδ>0sup{|y∗(Tx)|:y∗∈Y∗,x∈X,∥y∗∥=∥x∥=1,Rey∗(Gx)>1-δ} for every T∈ℒ⁢(X,Y).