投影算子之乘積及其相關問題

Abstract

本論文的目的是在探討那此希伯特空間上的算子，可分解成偏保距算子的乘積，以及 那些算子可分解成 (垂直)^投影算子的乘積。在有限維空間，算子T 是K 個偏保距算 子之乘積的充要條件是T 為收縮算子，而且rank(1-T*T)≦k•nullity T， 由此可推 論任一n 階不可逆收縮方陣皆可分解成n 個偏保距算子之乘積，而n 是所需的因子個 數的最小者。另一方面，在有限維空間，算子T 是有限個扱影算子之乘積扱影算子之 乘積的充要條件是T 么正等價於一個單位算子與一個不可逆的嚴格收縮算子的直和， 然而其所需的因子個數可任意。大在無窮維空間，我們得到有關於算子可分解成投影 算子之乘積的一些必要條件與充分條件，並且解決了自伴算子的分解問題。此外，我 們證明了一個嚴格收縮算子與一箇無窮維的零算子的直和可分解成有限個投影算子之 乘積。 /////// In this theisi, we consider bounded operators, on a complex separable Hilbert space, which are expressible as a product of partial isometries or orthogonal projections. More precisely, we show that a finite matrix T is the product of k partial isometries (k≧1) if and only if T is a contraction (∥T∥≦1) and rank (1-T*T) ≦k•nullity T. It follows, as a corlooary, that any n*n singular contraction is the product of n partial isometries and n is the smallest such number. On the other hand, a matrix T is the product of finitely many orthogonal projections if and only if T is unitarily equivalent to 1 ♁ S where S is a singular strict contraction (∥S∥<1). As contrasted to the previous case, the number of factors can be arbitrarily large. In addition, we obtain some necessary/sufficient conditions for bounded operators, on infinite-dimensional spaces, expressible as a product of projections, and we solve completely this problem for hermitian operators. Among other things, we show that an operator of form S ♁ 0, where ∥S∥<1 and the zero operator 0 acts on an infinite-dimensional space, is a product of finitely many projections. Finally, we give some related confectures and open problems.