Representation theorems for sesquilinear forms

2017. 10. 19. 14:00
Rosario Corso (Università degli Studi di Palermo)

Given a bounded sesquilinear form $\Omega$ defined on a Hilbert space $(\mathcal{H},\langle \cdot|\cdot \rangle)$, a basic result establishes the representation $\Omega(\xi,\eta)=\langle T\xi | \eta \rangle,  \forall \xi,\eta\in  \mathcal{H}$ with a unique bounded linear operator $T$. In the unbounded case one typically consider a sesquilinear form defined on a dense domain and try to find an (unbounded) operator with the same type of representation. It is not always possible to get an operator with the same domain of the form, but some conditions allow to obtain the properties of being densely defined and closed. Kato was one of the pioneers of this topic. In this talk we discuss his representation theorems, their applications and some generalizations.