Operator Projective Line and Its Transformations

We introduce a concept of the operator (non-commutative) projective line \(P\mathsf {H}{}\) defined by a Hilbert space \(\mathsf {H}{}\) and a symplectic structure on it. Points of \(P\mathsf {H}{}\) are Lagrangian subspaces of \(\mathsf {H}{}\) . If a particular Lagrangian subspace is fixed then we can define \(\mathrm {SL}_{2}(\mathbb {R})\) -action on \(P\mathsf {H}{}\) . This gives a consistent framework for linear fractional transformations of operators. Some connections with spectral theory are outline as well.