Alejandro Vareja
The Riemann Sphere of Operators on Hilbert Space
Given a Hilbert space $H$, consider $$ \tilde a= \begin{pmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{pmatrix} \in B(H\oplus H) $$ (with $a_{ij}\in B(H)$). The Riemann sphere of $B(H\oplus H)$ is the unitary orbit \[ \mathcal{R} = \left\{ \tilde u \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix} \tilde u^{-1} :\, \tilde u \in \mathcal{U}(H\oplus H) \right\}, \] where $\mathcal{U}(H\oplus H)$ denotes the unitary operators in $B(H\oplus H)$. We present the elements of $\mathcal{R}$, which are given by projections $P_{\mathrm{Gr}(T)}$ onto the graphs $ \mathrm{Gr}(T)=\{(h,Th): h\in H\} $ of densely defined closed operators $T$. We discuss general results and concrete examples, such as the case of the unbounded operator $-i\frac{d}{dx}$. In this context, we exhibit geodesics of projections onto graphs, natural approximations of unbounded operators by bounded operators, and Jacobi fields related to Fredholm operators.
