Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result:

**Theorem 1.** Let $p \in [1,\infty]$. The following assertions are equivalent:

(i) $T^n$ converges to $0$ with respect to the operator norm as $n \to \infty$ (in other words: the spectral radius $r(T)$ of $T$ satisfies $r(T) < 1$),

(ii) For each bounded sequence $u = (u_n)_{n \in \mathbb{N}_0} \subseteq X$ the convolution $ \left( \sum_{k=0}^n T^k u_{n-k} \right)_{n \in \mathbb{N}_0} \subseteq X $ is a bounded sequence, too.

(iii) For each sequence $(u_n)_{n \in \mathbb{N}_0} \subseteq X$ that converges to $0$, the convolution $ \left( \sum_{k=0}^n T^k u_{n-k} \right)_{n \in \mathbb{N}_0} \subseteq X $ converges to $0$, too.

(iv) For each sequence $u \in \ell^p(\mathbb{N}_0;X)$ the convolution $ \left( \sum_{k=0}^n T^k u_{n-k} \right)_{n \in \mathbb{N}_0} $ is in $\ell^p(\mathbb{N}_0;X)$, too.

For the continuous time case (i.e., where the powers $(T^n)_{n \in \mathbb{N}_0}$ are replaced with a $C_0$-semigroup on $X$), this theorem - along with various further conditions - can be found in Theorem 5.1.2 of the book "*Arendt, Batty, Hieber, Neubrander: Vector-Valued Laplace Transforms and Cauchy Problems (Birkhäuser, 2011)*". It is not difficult to see that the same proof works in the discrete time setting, too; so Theorem 1 is indeed true. However, I do not know an explicit reference for this result.

**Background.**

A colleague and I would like to state Theorem 1 as background material in an article. But we would prefer a reference instead of (a) including a complete proof and instead of (b) just claiming that "the proof works similarly as in the continuous time setting that is treated in [op. cit., Theorem 5.1.2]".

I find it hard to believe that such a theorem can only be found explicitly in the literature in the continuous time setting, but not in the discete time setting although the latter is typically easier to handle in the infinite-dimensional case.

On the other hand, the notes at the end of [op. cit., Chapter 5] indicate that the equivalent assertions in the continuous time result in [op. cit., Theorem 5.1.2] have actually been collected from various different articles, so it might well be that [op. cit., Theorem 5.1.2] is the first place in the literature where this result can be found in a consolidated form. So it

*does*at least seem possible that nobody has done the same work for the discrete time setting, yet.I've asked the same question to one of the authors of [op. cit.], who did not know a reference for the discrete time setting, though.