Sesquilinear forms as eigenvectors in quasi *-algebras, with an application to ladder elements

Abstract

We consider a particular class of sesquilinear forms on a Banach quasi *-algebra $\left(A\right[\parallel .\parallel ],A_0{[\parallel .\parallel }_0\left]\right)$ that we call eigenstates of an element $a\in A$, and we deduce some of their properties. We further apply our definition to a family of ladder elements, that is, elements of $A$ obeying certain commutation relations physically motivated, and we discuss several results, including orthogonality and biorthogonality of the forms, via Gelfand–Naimark–Segal (GNS) representation.