Simultaneous Williamson's normal form for positive operators

Abstract

We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A,B$ on a real symplectic Hilbert space $(\mathbf{H},J)$. Under a natural commutativity condition, there exists a (generally unbounded) operator $\Phi$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A={\Phi }^t{M}_A\Phi$, $B={\Phi }^t{M}_B\Phi$, and maps $J$ to a canonical symplectic operator. The result generalizes Williamson's normal form to infinite dimensions and yields a complete spectral invariant for the pair $(A,B)$.

arXiv comment19 pages