A Family of Semi-norms Between the Numerical Radius and the Operator Norm

Abstract

This paper introduces a new family of non-negative real-valued functions on the algebra of all bounded linear operators on a Hilbert space. These functions will be called semi-norms because they satisfy the norm axioms, except for the triangle inequality, which will be discussed for special cases. Many bounds and relations will be shown for this new family, with a connection to the existing literature.