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

We consider a particular class of sesquilinear forms on a Banach quasi *-algebra (A[parallel to.parallel to],A0[parallel to.parallel to 0])$({\cal A}[\Vert .\Vert],{\cal A}_0[\Vert .\Vert _0])$ that we call eigenstates of an element a is an element of A$a\in {\cal A}$, and we deduce some of their properties. We further apply our definition to a family of ladder elements, that is, elements of A${\cal A}$ obeying certain commutation relations physically motivated, and we discuss several results, including orthogonality and biorthogonality of the forms, via Gelfand-Naimark-Segal (GNS) representation.