國立清華大學 數學系

Abstract This talk represents some of our initial efforts to develop what we call a non-selfadjoint version of random operator theory. It is well known that, on finite dimensional vector spaces, random matrix theory has evolved into a remarkably sophisticated subject. On infinite dimensional spaces, most works on random operators so far are restricted to the selfadjoint case, such as random Schrodinger operators. A non-selfadjoint theory is largely missing so far. We seek to understand the random counterpart of a canonical non-selfadjoint operator, namely, the unilateral shift, defined as $$Te_n=e_{n+1}, \quad n=1, 2, \cdots,$$ where $\{e_n\}_{n=1}^\infty$ is an orthonormal basis for a separable complex Hilbert space. This seemingly naive operator encompasses a broad range of beautiful theorems, especially in its connection to complex analysis. Our goal is to consider its random counterpart: $$Te_n=X_ne_{n+1}, \quad n=1, 2, \cdots,$$ where $\{X_n\}_{n=1}^\infty$ is a sequence of i.i.d. random variables. This clearly fundamental model seems to elude investigation in the literature so far.