Let $K$ be a compact connected subset of the complex plane, of non-void interior, and whose complement in the extended complex plane is connected. Denote by $F_n$ the $n$-th Faber polynomial associated with $K$. The aim of this paper is to find suitable Banach spaces of complex sequences, $\mathcal{R}$, such that statements of the following type hold true: if $T$ is a bounded linear operator acting on the Banach space $\mathcal{X}$ such that $( \langle F_n(T)x,x^\ast \rangle )_{n\ge 0} \in \mathcal{R}$ for each pair $(x,x^{\ast}) \in \mathcal{X}\times \mathcal{X}^{\ast}$, then the spectrum of $T$ is included in the interior of $K$. Generalisations of some results due to W. Mlak, N. Nikolski and J. van Neerven are thus obtained and several illustrating examples are given. An interesting feature of these generalisations is the influence of the geometry of $K$ and the regularity of its boundary.
