Sup and Max Properties for the Numerical Radius of Operators in Banach Spaces

The article deals with the following problem: given a bounded linear operator A in a Banach space X, how can multiplication of A by an operator of norm one (contraction) affect the numerical radius of A? The approach used in this work is close to that employed by Vieira and Kubrusly in 2007 for their study concerning spectral radius. It turns out that this study is closely related to the study of V-operators conducted in 2005 by Khatskevich, Ostrovskii, and Shulman; the results of this article demonstrate that in certain cases the obtained property of an operator implies that it is a V-operator, while in some other cases the converse is true.