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arXiv · 2610.04583

Simultaneous Williamson's normal form for positive operators

Rudra Kamat·Anmary Tonny
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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 $Φ$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A = Φ^t M_A Φ$, $B = Φ^t M_B Φ$, 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 $(A,B)$.

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Rudra Kamat, Anmary Tonny. 2026-10-03. Simultaneous Williamson's normal form for positive operators. https://arxiv.org/abs/2610.04583

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