Random Weighted Shifts(2/2)

We roughly divide linear analysis into two categories: on finitedimensional spaces (Euclidean spaces) and infinite dimensionalspaces (Banach spaces/Hilbert spaces, eg.). On a finitedimensional space, we have the familiar matrix theory. On aninfinite dimensional space, we usually call it “Operator Theory”.Of course, these two areas are largely different in methodology.If we ask the entries in a matrix to be random variables, then wehave the so-called random matrices. This has developed into adeep area in mathematics. On infinite dimensional spaces,however, the random theory is poorly understood so far. Ofcourse, for some special cases, such as random Schrodingeroperators, due to their physical background, are carefullyanalyzed. Random Toeplitz operators are also studied, but notsystematically. So far, as far as the general operator theory isconcerned, the random theory is largely untapped. This demandsa decent understanding of both operator theory and probability,and one needs to find the right questions to ask. In this proposal,we attempt to start with the most important non-self-adjointoperator：the unilateral shift, and develop a random theory for it.Specifically, we will define a random weighted shift model, andtry to carry the classical theory over to this random object in aparallel way. Of course, most of the time, this carry-over resultsin triviality. But we do find some non-trivial problems/results.