I'm learning about model spaces
Note: this was originally a thread on my twitter . Last time I talked about subspaces of the Hardy space that are invariant under the shift operator . Now let's talk about the backward shift! (Most of the discussion here comes from the excellent "Introduction to Model Spaces and their Operators" by Garcia, Mashreghi, and Ross, which I cannot recommend highly enough.) Our shift invariant subspaces are of the form \(uH^2\) where u is an inner function The backward shift is the adjoint of the shift, so in some sense, invariant subspaces for the backward shift should be "opposite" of those of the shift. In fact, we define "model spaces" as the orthogonal complement in \(H^2\) of \(uH^2\)! As a reminder of your linear algebra, the orthogonal complement of a subspace is everything orthogonal to every element of that subspace. That is, if you take the inner product of anything in a subspace V with anything in the orthogonal complement V-perp, you get zero I...