This is the first post of a series of posts in which we will eventually venture deep into the realm of coarse geometry. But we will always be motivated by questions which are related to the one that we will discuss here.
But our first steps into coarse geometry will be very gently: we will be concerned with infinite matrices and their operator norms. We will see that if they are not band diagonal, then it becomes much more difficult to estimate their norms.
Norms of band matrices
We consider an infinite band matrix \(T\) whose rows and columns are indexed by the integers \(\mathbb{Z}\). This means that \(T = (T_{m,n})_{m,n \in \mathbb{Z}}\) satisfies \[T_{m,n} = 0 \text{ for all } m,n \text{ with } |m-n| > N\] for some \(N \in \mathbb{N}\). If the entries of \(T\) are uniformly bounded, i.e., \[\sup_{m,n} |T_{m,n}| < K\] for some \(K \in \mathbb{R}\), then it is straight-forward to show that \(T\) acts continuously on bi-infinite vectors from \(\ell^2(\mathbb{Z})\) and the operator norm of \(T\) can be estimated from above as \[\|T\| \le 2N \cdot K.\] (The factor of \(2\) comes from the fact that the thickness of the band of non-zero entries of \(T\) is \(2N\).)
But if \(T\) is not band diagonal, how to estimate its operator norm (in terms of its entries)?
Small entries, but big norm
To show how complicated the above question is, let us consider the following related problem: we are given an infinite matrix \(T\) and we are told that it is in the closure (in operator norm) of the band matrices with uniformly bounded entries. How can we then actually construct a sequence of band matrices \(T^{(R)}\) with \(T^{(R)} \to T\)?
A natural first approach would be the following (at least, it was my first approach): we cut down \(T\) along thick diagonals and then try to show that the resulting sequence of band matrices actually approximates \(T\). Concretely, writing \(T = (T_{m,n})_{m,n \in \mathbb{Z}}\) and fixing \(R \in \mathbb{N}\), we define a band matrix \(T^{(R)}\) by \[T^{(R)}_{m,n} := \begin{cases} T_{m,n} & \text{ if } |m-n| \le R\\ 0 & \text{ otherwise }\end{cases}\] The hope is that \(T^{(R)} \to T\) as \(R \to \infty\).
But the above approach is doomed to fail: it can even happen that \(\|T^{(R)}\| \to \infty\)!
So even if \(T\) is in the closure (in operator norm) of the band matrices, it does not mean that the terms off a thick-diagonal of \(T\) define an infinite matrix of small norm.
An exercise for you
Now it is your turn – try to figure out how to solve the problem from the previous section, i.e., given an infinite matrix in the closure of the band matrices, how to construct a corresponding approximating sequence?
Outlook
The above question about operator norms of infinite matrices is the starting point of our journey through coarse geometry. In a future post we will do our next step and address the following two questions:
- How to determine whether \(T\) is approximable (in operator norm) by band matrices?
- Provided \(T\) is approximable by band matrices, how to actually construct an approximating sequence?
Generalizing from bi-infinite vectors indexed by \(\mathbb{Z}\) to vectors indexed by a metric space \(X\) and from band matrices to so-called operators of finite propagation, we will see that (the answers to) the above questions lead us to a 30-year-old conjecture of John Roe and to coarse geometric notions like asymptotic dimension.