This project is mathematical research in operator theory, a subject that had its origins in quantum mechanics (where operators on Hilbert space are used in place of scalars to represent physical quantities), but that has long since acquired a vigorous life of its own. Operators interact with one another in a much richer way than do scalars, and individual operators carry a great deal of structure (spectrum, numerical invariants such as index, and so forth) whose elucidation is a perpetual challenge to the subject. Part of Professor Fialkow's research deals with spectrum and index for so-called elementary operators, which are linear combinations of left and right multiplications. The main thrust here is to recover information about the given elementary operator in terms of multivariable spectral data from the pieces of which it is composed. Another aspect has to do with quasisimilarity, two operators being quasisimilar if they are intertwined left and right by a pair of injective transformations with dense range; the objective is to characterize the operators that are quasimilar to the unilateral shift.
