关于完全不可约算子

On Completely Irreducible Operators

本文简要总结了吉林大学泛函分析讨论班十多年来关于完全不可约算子的工作。全文共分四节。第1节详细阐述了完全不可约算子的背景。第2节展示出一些熟知的算子类,如加权移位,Toeplitz算子等,其中哪些算子是完全不可约的。第3节证明Hilbert空间上每个有界线性算子都可以用完全不可约算子的有限直接和来逼近。从而证实,完全不可约算子是Jordan块比较合适的类似物。第4节讨论与完全不可约算子有关的问题,显示完全不可约算子的某些性质是不同于单胞算子的。

Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H=M+N and M∩N={0}, M is said to be a completely reduced (?)bspace of T. If T has a completely reduced subspace which is nontrivial, T is said to be completely reducible; otherwise, T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that have been done by functional analysis seminar of Jilin University in the past more than ten years.The paper contains four sections. In section 1 the background of completely irreducible operators is detailed. Section 2 shows which operator in some well known classes of operators, for example, weighted shifts, Toeplitz operators, etc. , is completely irreducible. In section 3 it is proved that every bounded linear operator on Hilbert space can be approximated by the finite direct sum of completely irreducible operators. It is clear that completely irreducible operator is a rather suitable analogue of Jordan block in L (H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties of completely irreducible operators are different from those of unicellular operators.##原图像不清晰