Problems in Random Operator Theory(4/4)

We divide linear analysis roughly into two categories: on finite dimensional spaces (Euclidean spaces) and infinite dimensional spaces (Banach spaces/Hilbert spaces, e.g.). On a finite dimensional space, we have the familiar "Matrix Theory". On an infinite 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 we have the so-called random matrices. This has developed into a deep area in mathematics. On infinite dimensional spaces, however, the random theory is poorly understood so far. Of course, for some special cases, such as random Schrodinger operators, due to their physical background, are carefully analyzed. Random Toeplitz operators are also scarely studied, but far from being systematic. Currently, as far as the general operator theory is concerned, the random theory is essentially untapped. This demands a good understanding of both operator theory and probability theory, and, moreover, one needs to find the right questions to ask. In this proposal, we attempt to start with the most important non-self-adjoint operator：the unilateral shift, and develop a random theory for it. Then the second problem we consider will be the connection between point processes and random zero sets of analytic functions.