谱连通的双拟三角算子+小紧=τ(N)∩(SI)

Biquasitriangular+small compact operators=τ(N)∩(SI)

设N是连续套,τ(N)={T∈L(H),TMM;M∈N}.纪友清[5]等人得出:连续套代数中强不可约算子酉轨道闭包是全体谱连通的双拟三角算子。此外蒋春澜等[6]证明了若T是谱连通的双拟三角算子,ε>0,则存在紧算子K,‖K‖<ε,使得T+K∈(SI).利用文献[2]中定理2.3.1证明了连续套代数中强不可约算子酉轨道闭包与全体谱连通的双拟三角算子集合恰好相差小范数紧算子,即:谱连通的双拟三角算子+小紧=τ(N)∩(SI).

Let N denote continuous nest, and let τ（N）={T∈L（H）,TMM;M∈N}Ji Y Q [5] have proved that the closure of the unitary orbit of the strongly irreducible operators in continuous nest algebras is equal to the set of all biquasitriangular operators whose spectrum is connected. Moreover, Jiang C L[6 ] have proved that, if T is a biquasitriangular operator whose spectrum is connected, then for given ε〉0 ,there exists a compact operator K such that ｜｜K｜｜〈εandT＋K∈（SI） By mean of [2, Theorem 2.3.1 ] , it is shown that there are small compact operators between the closure of the unitary orbit of the strongly irreducible operators in a continuous nest algebra and the set of all biquasitriangular operators whose spectrum is connected. That is, biquasitriangular operator of ＋ small compact operators =τ（N）∩（SI）