Boundedness and compactness of composition operators
I’m not an expert on composition operators and I’ve never published a paper on this subject — but I do have an opinion.
For some reason I keep on getting referee requests and requests to write Math Reviews for papers on boundedness and compactness of composition operators between all sorts of non-standard and perhaps exotic spaces (Bloch, Bloch-type, Besov, Q_p, Q_K, etc…). I find these papers tedious. There don’t seem to be any new ideas here. On the Hardy space, boundedness comes essentially for free via Littlewood. Compactness is fascinating due to the Nevanlinna counting function – beautiful stuff. Computing the essential norm makes a fascinating link to Clark measures.
For other spaces, boundedness and compactness gets tricky and some people (some time ago) have worked out some nice results along these lines for the more standard spaces such as the Bergman and Dirichlet spaces and their close cousins. Beyond these spaces, I think boundedness/compactness of composition operators becomes rather dry since the answer is nearly the same as are the techniques to prove them. Plus, these problems seem to have run their course and do not have the broad appeal they once did. I’m definitely not alone with this opinion. I often sit next to people at conferences who also feel the same way.
Composition operators is a wonderful subject with a proud history and a great future. For example, there is some recent work of Jury, Kriete, MacCluer, Moorehouse, Quertermous, etc., dealing with algebras generated by composition operators which makes links to other areas of analysis and even algebra. Why not work on these types of problems? In addition, this is all on the good old Hardy space H^2. What are the invariant subspaces of a fixed composition operator? There are several nice papers on this. Cyclic vectors? Connected components of the space of composition operators? What about the dynamics of composition operators – hypercyclic, supercyclic, etc? What about the spectrum? These are all interesting composition operator problems which all take place on H^2 (perhaps even the Bergman space to spice things up a bit) and will be interesting to a wider operator theory audience since these are natural questions one asks about any operator on a Hilbert space. These problems also have the added bonus in that they connect the operator theory of the composition operator with the complex analysis of the symbol. The boundedness and connectedness problem on the exotic spaces certainly connect the symbol with the operator but not really the complex analysis of the symbol since the results are often sup and limsup conditions.
So, I realize I’m opening myself up to criticism but I think composition operators needs to stick closer to H^2 and not get into all the exotic spaces.
Thanks for reading.