We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form

The classical mean and pointwise ergodic theorems due to von Neumann and Birkhoff, respectively, take their origin in questions from statistical physics and found applications in quite different areas of mathematics such as number theory, stochastics, and harmonic analysis. Over the years, they were extended and generalised in many ways. For example, to multiple ergodic theorems, see Furstenberg [

The return time theorem due to Bourgain, solving a quite long standing open problem, is a classical example of a weighted pointwise ergodic theorem. It states that for every measure preserving system

The most general class of systems for which the convergence in the multiple return time theorem is known to hold

In this paper, we search for good weights for ergodic theorems using a functional analytic perspective and produce deterministic good weights. We first introduce sequences of the form

We finally remark that all results in this paper hold if we replace linear sequences by a larger class of “asymptotic nilsequences,” that is, for sequences

A linear operator

We call a sequence

A large class of operators with relatively weakly compact orbits, leading to a large class of linear sequences, are power bounded operators on reflexive Banach spaces. Recall that an operator

By restricting to the closed linear invariant subspace

We obtain the following structure result for linear sequences as a direct consequence of an extended Jacobs-Glicksberg-deLeeuw decomposition for operators with relatively weakly compact orbits.

Every linear sequence is a sum of an almost periodic sequence and a (bounded) sequence

Let

Let

In this section, we show that one can take linear sequences as weights in the multiple Wiener-Wintner type generalisation of the return time theorem due to Zorin-Kranich [

First we recall the definition of a property satisfied universally.

Let

We show the following linear version of the Wiener-Wintner type multiple return time theorem.

For every

By Theorem

Universal convergence for almost periodic sequences is a consequence of Zorin-Kranich’s result [

Using the Host-Kra Wiener-Wintner type result for nilsequences and extending their result for linear polynomials from [

The following result is a consequence of Chu [

Every nilsequence is a good weight for the multiple polynomial ergodic theorem.

This remains true when replacing a nilsequence by a linear sequence.

Every linear sequence is a good weight for the multiple polynomial ergodic theorem.

For an almost periodic sequence

The following example shows that if one does not assume relative weak compactness in the definition of linear sequences, each of the previous results can fail dramatically even for positive isometries on Banach lattices.

Let

In particular, for

We further show that in fact for every

Take

The author thanks Pavel Zorin-Kranich for helpful discussions.