**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.

- M.I. Arenas-Herrera, L. Verde-Star: Representation of doubly infinite matrices as non-commutative Laurent series,
*Spec. Matrices*, 5 (2017), 250-257. - 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. - R. Słowik: Some counterexamples for Cayley-Hamiltonian theorem for doubly infinite matrices, submitted.