Definition [Costakis, Manoussos, Parissis] Let $X$ be a Banach space. We say that a bounded linear operator $T\colon X\to X$ is

1. Recurrent: if for each $x\in X$ there exists an increasing sequence $(n_k)$ of natural numbers such that $T^{n_k}(x)$ converges to $x$.
2. Rigid: if there exists an increasing sequence of natural numbers $(n_k)$ such that $T^{n_k}(x)$ is convergent to $x$ for every $x\in X$.
3. Uniformly rigid: if there exists an increasing sequence $(n_k)$ of natural numbers such that $\lim_k |T^{n_k}-I|=0$.

In these consecutive talks, we report on joint work by M.J. Beltrán, E. Jordá and J. Galindo where we study the recurrence and rigidity of multipliers on semisimple Banach algebras and analyze the case of the Fourier algebra $A(G)$ of a locally compact group.

We will address the following problems:

– Let $T\colon \mathfrak{A}\to \mathfrak{A}$ be a multiplier of a semisimple Banach algebra $\mathfrak{A}$ with spectrum $\Delta(\mathfrak{A})$. Its Gelfand transform, $\widehat{T}$, then defines a multiplier $M_{_{\widehat{T}}}\,\colon C_0(\Delta(\mathfrak{A}))\to C_0(\Delta(\mathfrak{A}))$. Find the relations between the recurrence and rigidity of these multipliers and determine under which conditions they coincide.

– Describe the recurrent and (uniformly) rigid multipliers of the Fourier algebra and find examples separating these concepts.

Our results show that for power-bounded multipliers on the most common algebras (including Fourier algebras), the recurrence and rigidity properties of a multiplier $T$ coincide exactly with those of $M_{_{\widehat{T}}}$; this will be the subject of Talk I. In the absence of power-boundedness this equivalence fails, and we provide counterexamples within the framework of Fourier algebras; these will be discussed in Talk II.