全非零矩阵与全算子代数的生成元
Fully non-zero matrices and generators of full algebras of operators
浏览论文内容
中文总结 AI 辅助
本文推广了非纯量矩阵相似于全非零矩阵的结果并应用于算子生成问题，还证明了TVS子集线性无关性与TA生成元的稳定性。
中文摘要 AI 辅助
在众多领域中，每个非纯量矩阵M都相似于一个所有元素均非零的矩阵，我们对该结果进行了推广，包括除环情形下的推广。随后将这些结果应用于“哪些算子对可生成有限维和无限维空间上全体算子构成的代数”这一问题。对于复可分无限维希尔伯特空间，该问题与未解决的不变子空间问题密切相关。最后证明，任意拓扑向量空间（TVS）子集的线性无关性与任意拓扑代数（TA）的生成元在所提定理的意义下是稳定的。
英文摘要
In a wide variety of fields, every non-scalar matrix $M$ is similar to a matrix all of whose entries are non-zero. We give extensions of this result, including some in the division ring context. We then apply these results to the question of which pairs of operators generate the algebra of all operators on finite- and infinite-dimensional spaces. In the case of complex, separable infinite-dimensional Hilbert space, this is intimately related to the unsolved Invariant Subspace Problem. Finally, linear independence of subsets of arbitrary TVS's and generators of arbitrary TA's are shown to be stable in the sense of the theorems presented.
