| 摘要: | 我們可以大概把線性分析分在有限維空間上（即歐式空間） 與無窮維空間上（巴拿赫空間/希爾伯特空間）。在有限維空 間上，我們稱之為矩陣理論。在無窮維空間上，我們稱之為 算子理論。當然，這兩個分支的研究方法大相徑庭。 若我們讓矩陣的元素取成隨機變量，則我們得到所謂的隨機 矩陣。這已經發展成為數學中一門深刻的學問。大家輩出。 而無窮維空間上對應的隨機算子理論則甚少被理解。當然有 些特殊情況，如隨機薛定諤算子，因為其物理背景，被認真 的研究。隨機Toeplitz 算子也有一些（不系統）的研究。目 前，就一般算子理論而言，其隨機版本基本上是空白的。這 需要對抽象算子理論與機率論都有較深刻的理解，從而找到 合適的問題來切入。在這項計劃裡，我們試圖從最重要的非 自伴算子的隨機版本開始，建立一套新的，系統的理論。 具體而言，我們將定義一個新的隨機算子模型，並試圖將經 典算子理論平行搬到隨機版本上。當然，這種搬法大部分時 候都是平凡或無解的。但我們確實找到一些好的問題，同時 具有深度和可解性（或可能可解）。在這項計劃裡，我們報 告一些目前取得的部分結果，以及接下來準備探討的問題。 ;We roughly divide linear analysis into two categories: on finite dimensional spaces (Euclidean spaces) and infinite dimensional spaces (Banach spaces/Hilbert spaces, eg.). 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 studied, but not systematically. So far, as far as the general operator theory is concerned, the random theory is largely untapped. This demands a 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-adjoint operator：the unilateral shift, and develop a random theory for it. Specifically, we will define a random weighted shift model, and try to carry the classical theory over to this random object in a parallel way. Of course, most of the time, this carry-over results in triviality. But we do find some non-trivial problems/results. |
