A Collection of Seminorms Linking the A-numerical Radius and the Operator A-seminorm

We investigate a novel operator seminorm, QA,mλ,f, for an A-bounded operator Q, where A is a positive operator on a complex Hilbert space (K,⟨·,·⟩). This seminorm is defined using a continuous increasing and bijective function f:R+⟶R+ and an interpolational path mλ of the symmetric mean m. Specifically, QA,mλ,f=supf−1fQy,yAmλfQyA:y∈K,yA=1, where f−1 represents the reciprocal function of f, and ⟨·,·⟩A and ·A denote the semi-inner product and seminorm, respectively, induced by A on K. We explore various bounds and relationships associated with this new concept, establishing connections with existing literature.