6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Roksana Słowik

Doubly infinite matrices and the Cayley-Hamiltonian theorem


Let K be a doubly infinite matrix. If K is a polynomial in S and S-1, where by S we mean the shift matrix, then K is called a Laurent matrix. During the talk we will present some results connected to the following question.

If K is finite band and periodic (but not tridiagonal) is there a polynomial Q 
so that Q(K) is a Laurent matrix?

We will show that the answer to the above question, in general, is no. We will also provide some examples and further directions to study this issue.

  1. M.I. Arenas-Herrera, L. Verde-Star: Representation of doubly infinite matrices as non-commutative Laurent series, Spec. Matrices, 5 (2017), 250-257.
  2. B. Simon: A Cayley-Hamilton theorem for periodic finite band matrices,Functional analysis and operator theory for quantum physics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2017, 525-529.
  3. R. Słowik: Some counterexamples for Cayley-Hamiltonian theorem for doubly infinite matrices, submitted.
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