In the first post of this series we asked at the end two questions – in this post we start working towards the answers in the general setup of the third post of this series.
Our setup from the third post is the following: We have a metric space \((X,d)\) and we consider a bounded, linear operator \(T\) acting on \(\ell^2(X)\). We further assume that \((X,d)\) has bounded geometry – as explained in the third post of this series, this implies that every linear operator on \(\ell^2(X)\) of finite propagation and with uniformly bounded entries will be bounded.
The first question from the first post was, phrased in the above language, how to determine whether \(T\) is in the closure (in operator norm) of the bounded, linear operators on \(\ell^2(X)\) of finite propagation?
To answer it, we introduce the following notion: a bounded, linear operator \(T\) acting on \(\ell^2(X)\) is called quasi-local, if there exists a function \(\mu\colon \mathbb{R}_{> 0} \to \mathbb{R}_{\ge 0}\) with \(\mu(R) \to 0\) as \(R \to \infty\) such that for all vectors \(v \in \ell^2(X)\) we have \[\|T v\|_{X \setminus \mathrm{Neigh}(\mathrm{supp}(v),R)} \le \mu(R) \cdot \|v\|\,,\] where we have used the following notions:
- \(\mathrm{supp}(v)\) is the support of the vector \(v\).
- \(\mathrm{Neigh}(\mathrm{supp}(v),R)\) is the \(R\)-neighbourhood of the support of \(v\), i.e., \[\mathrm{Neigh}(\mathrm{supp}(v),R) := \{x \in X\colon d(x,\mathrm{supp}(v)) \le R\}\,.\]
- \(\|w\|_{K}\) for a subset \(K \subset X\) and for a vector \(w \in \ell^2(X)\) is the (semi-)norm defined as the square-root of \(\sum_{x \in K} |w(x)|^2\).
It is clear that if \(T\) has finite propagation, then \(T\) is quasi-local: in this case we can choose \[\mu(R) := \begin{cases} \|T\|& \text{for }R \le \mathrm{prop}(T)\,,\\0& \text{for }R > \mathrm{prop}(T)\,.\end{cases}\] Here \(\mathrm{prop}(T)\) denotes the propagation of \(T\).
Let us reformulate the quasi-locality condition a bit. If \(f\) is a bounded function on \(X\), then we can define the multiplication operator \(M_f\) acting on \(\ell^2(X)\) as follows: \[(M_f v)(x) := f(x)v(x)\,.\] It is a bounded, linear operator with \(\|M_f\| = \|f\|_{\infty}\). It is now straight-forward to show that a bounded, linear operator \(T\) acting on \(\ell^2(X)\) is quasi-local if and only if for every \(\varepsilon > 0\) exists \(R > 0\) such that for any two bounded functions \(f,f^\prime\) on \(X\) whose supports are separated by a distance of at least \(R\) we have the operator norm estimate \(\|M_f T M_{f^\prime}\| \le \varepsilon \cdot \|M_f\| \cdot \|M_{f^\prime}\|\).
Using this reformulation one can easily verify that the condition of being quasi-local is closed in operator norm, i.e., if \((T_n)_{n \in \mathbb{N}}\) is a sequence of quasi-local operators converging in operator norm to \(T\), then \(T\) will also be quasi-local.
The upshot of all the above is the following: we have a condition on bounded, linear operators, namely being quasi-local, which is shared by all operators of finite propagation and which is closed under the operator norm. And actually, to me quasi-locality exactly looks like the norm closure of the property of having finite propagation, i.e., it might be the sought answer to the first question of the first post of this series.
So the new question is now: if \(T\) is quasi-local, is it approximable by operators of finite propagation? To answer it we would have to come up with a procedure to extract an approximating sequence of operators of finite propagation for \(T\) – which means solving the second question of the first post of this series.
Note that the naive approach of finding an approximating sequence of operators of finite propagation, namely to cut down the matrix representative \((T_{x,y})_{x,y \in X}\) of \(T\), where \(T_{x,y} := (T\delta_y)(x)\) for the Dirac vector \(\delta_y \in \ell^2(X)\), to a band matrix, is doomed to fail as the example from the second post of this series shows. So we have to come up with something different. And this will be explained in the next post in this series!