A Study on Unbounded Operators related to Quantum Groups

与量子群相关的无界算子研究

基本信息

批准号：
12640184
负责人：
OTA Schoichi
金额：
$ 0.96万
依托单位：
Kyshu Institute of Design
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

Motivated by recent developments in the theory of quantum groups, some classes of q-deformed operators (q-normal, q-quasinormal, q-deformed hyponormal operators) are investigated systematically, where q is a positive deformation parameter.1. In the case when 0<q<1, a non-trivial q-deformed hyponormal closed operator must be unbounded. Its spectrum contains zero, the point spectrum is either {0} or empty set and its planar Lebesgue measure is always positive. If a q-deformed hyponormal closed operator T has dense range, then the inverse is also q-deformed hyponomal and the real and imaginary parts of its inverse are presented by those of T intertwined with some pure contraction.2. On the other hand, if q>1 then there exists a q-deformed hyponormal operator with empty spectrum, which is constructed by an unbounded weighted shift, though the spectrum of every bounded q-qusinormal operator consists only of zero.3. It is known that the Fuglede-Putnam theorem for usual normal operators is one of the most useful results in operator theory. Related to this, a q-analogue of the Fuglede-Putnam theorem for q-normal operators is proposed and is characterized in terms of the attached contraction.4. The spectral analysis for q-deformed operators seems to be difficult. For example, the spectrum of all q-normal weighted shift is the whole complex plain. To analyze for such operators, a certain operator matrix representation induced by a closed operator is introduced and studied.

由量子群理论的最新发展激励，一些类别的Q形式的运算符（Q-正态，Q- Quasinormer，Q构成Q构造的不正常算子）被系统地研究，其中Q是一个正变形参数。1。在0 <q <1的情况下，必须无限制地无限制的非平凡Q构造的不正常闭合操作员。它的频谱包含零，点频谱是{0}或空集，其平面Lebesgue度量始终为正。如果Q构造的不正常的闭合算子T具有密度范围，则逆量子也是Q构造的拟态型 - 其反向的真实和虚构部分由T与某些纯收缩交织在一起的T表示。2。另一方面，如果Q> 1，则存在一个带有空谱的Q构造的士不良操作员，该操作员由无限的加权移动构建，尽管每个有界Q-Q-Qusinormoral oberator的光谱仅由零组成。3。众所周知，通常正常运算符的fuglede-putnam定理是操作者理论中最有用的结果之一。与此相关的是，提出了针对Q-正常运算符的Fuglede-Putnam定理的Q-Analogue，并根据附着的收缩为特征。4。 Q构造的操作员的频谱分析似乎很困难。例如，所有Q-正常加权移动的光谱是整个复合物平原。为了分析此类操作员，引入并研究了由封闭操作员引起的某个操作员矩阵表示。