论文标题
分析性，排名一个扰动和左光谱的不变性
Analyticity, rank one perturbations and the invariance of the left spectrum
论文作者
论文摘要
我们解决了分析运算符的等级第一扰动的分析性问题。如果$ \ mathscr m_z $是功能性的Hilbert Space $ \ Mathscrh_κ$和$ f \ in \ Mathscr H $的乘法运算符，则$ f（0）= 0，$ \ then $ \ Mathsscr m_z + f \ otimes 1 $始终分析。如果$ f（0）\ neq 0，$，则$ \ mathscr m_z + f \ otimes 1 $的分析性是根据成员的成员资格来特征的一种扰动。 In particular, we show that the left spectrum $σ_l(T + f \otimes g)$ of the rank one perturbation $T + f \otimes g,$ $\,g \in \ker(T^*),$ of a cyclic analytic left invertible bounded linear operator $T$ coincides with the left spectrum of $T$ except the point $ \ inp {f} {g}。$一般而言，点$ \ inp {f} {g} {g} $可能或不属于$σ_l（t + f \ otimes g）。但是，如果它属于$σ_l（t + f \ otimes g f \ otimies g backslash ipplash iplaime of phackslash ipl of plastslave） \ otimes g。$
We address the question of the analyticity of a rank one perturbation of an analytic operator. If $\mathscr M_z$ is the bounded operator of multiplication by $z$ on a functional Hilbert space $\mathscr H_κ$ and $f \in \mathscr H$ with $f(0)=0,$ then $\mathscr M_z + f \otimes 1$ is always analytic. If $f(0) \neq 0,$ then the analyticity of $\mathscr M_z + f \otimes 1$ is characterized in terms of the membership to $\mathscr H_κ$ of the formal power series obtained by multiplying $f(z)$ by $\frac{1}{f(0)-z}.$ As an application, we discuss the problem of the invariance of the left spectrum under rank one perturbation. In particular, we show that the left spectrum $σ_l(T + f \otimes g)$ of the rank one perturbation $T + f \otimes g,$ $\,g \in \ker(T^*),$ of a cyclic analytic left invertible bounded linear operator $T$ coincides with the left spectrum of $T$ except the point $\inp{f}{g}.$ In general, the point $\inp{f}{g}$ may or may not belong to $σ_l(T + f \otimes g).$ However, if it belongs to $σ_l(T + f \otimes g) \backslash \{0\},$ then it is a simple eigenvalue of $T + f \otimes g.$
