Faber polynomials of matrices for non-convex sets †

It has been recently shown that ||Fn(A)|| ≤ 2, where A is a linear continuous operator acting on a Hilbert space and Fn is the Faber polynomial of degree n corresponding to some convex compact E ⊂ C containing the numerical range of A. Such an inequality is useful in numerical linear algebra, it allows for instance to derive error bounds for Krylov subspace methods. In the present paper we extend this result to not necessarily convex sets E.