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} $(\text{A}[\mathrm{\parallel }\mathrm{.}\mathrm{\parallel }],\text{Ao}[\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_0])$ which we call {\em eigenstates of an element} $a\in \text{A}$, and we deduce some of their properties. We further apply our definition to a family of ladder elements, i.e. elements of $\text{A}$ obeying certain commutation relations physically motivated, and we discuss several results, including orthogonality and biorthogonality of the forms, via GNS-representation.