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PreprintCC BY 4.0arXiv

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 (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 = Φt MA Φ, B = Φt MB 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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Cite

BibTeXRIS
BibTeX
@misc{kamat2026simultaneous,
  title = {{Simultaneous Williamson's normal form for positive operators}},
  author = {Rudra Kamat and Anmary Tonny},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04583},
  eprint = {2610.04583},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04583},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Simultaneous Williamson's normal form for positive operators
AU  - Rudra Kamat
AU  - Anmary Tonny
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04583
UR  - https://arxiv.org/abs/2610.04583
AB  - We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A, B$ on a real symplectic Hilbert space (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).
ER  -