Sesquilinear forms as eigenvectors in quasi *-algebras, with an application to ladder elements
Fabio Bagarello, Hiroshi Inoue, Salvatore Triolo
TL;DR
The paper extends the notion of eigenvectors to sesquilinear forms on Banach quasi *-algebras by introducing eigenstates $\varphi$ of an element $a$ through $\varphi(b,a)=\lambda\varphi(b,e)$. It proves that if $a=a^*$ then $\lambda$ is real, and shows how the conjugate form $\psi(a,b)=\varphi(b^*,a^*)$ becomes an eigenstate of $a^*$ with eigenvalue $\overline{\lambda}$, linking left/right eigenstate notions. The authors then apply this framework to quon-like ladder elements, constructing a family $\varphi_l$ satisfying $\varphi_l(a,n_0)=\beta_l\varphi_l(a,e)$ with a recursive $\beta_l$ and introducing almost-coherent forms $\Phi_z$ that obey $\Phi_z(a x_0,b x_0)=z\Phi_z(a,b)$. Using a GNS construction, all $\varphi_l$ derive from a single representation with vectors $\xi_l=\pi_0(y_0)^l\xi_0$, yielding orthogonality when $y_0=x_0^*$ and a potential biorthogonal structure otherwise. Overall, the work opens avenues for coherent-form generalizations and deeper orthogonality analyses within the quasi *-algebra setting.
Abstract
We consider a particular class of sesquilinear forms on a {Banach quasi *-algebra} $(\A[\|.\|],\Ao[\|.\|_0])$ which we call {\em 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, i.e. elements of $\A$ obeying certain commutation relations physically motivated, and we discuss several results, including orthogonality and biorthogonality of the forms, via GNS-representation.