### Coarse embeddings and non-positive curvature

Let $$(X,d)$$ be a complete, geodesic metric space. $$(X,d)$$ is called an Alexandrov space of global non-positive curvature if for every quadruple of points $$x,y,z,w$$ such that $$w$$ is a metric midpoint of $$x$$ and $$y$$, i.e., $$d(w,x) = d(w,y) = d(x,y)/2$$, we have $d(z,w)^2 + d(x,y)^2/4 \le d(z,x)^2/2 + d(z,y)^2/2.$ If the reverse inequality … Continue reading "Coarse embeddings and non-positive curvature"

### Operators of finite propagation

In the starting post of this series we considered infinite band matrices (with uniformly bounded entries) acting on infinite vectors and asked at the end the question how to determine whether a given matrix, which is not a band matrix, can be approximated by such. Today we provide the setup in order to answer this question properly in … Continue reading "Operators of finite propagation"

### Equivariant band matrices and Fourier series

Recall that in the first post of this series we claimed that there exists an infinite matrix $$T$$ which is in the closure (in operator norm) of the band matrices with uniformly bounded entries, but for which we have $$\|T^{(R)}\| \to \infty$$. Here \[T^{(R)}_{m,n} := \begin{cases} T_{m,n} & \text{ if } |m-n| \le R\\ 0 & … Continue reading "Equivariant band matrices and Fourier series"

### Norms of infinite matrices

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 … Continue reading "Norms of infinite matrices"