论文标题
排名一度扰动的剩余性
Left-invertibility of rank-one perturbations
论文作者
论文摘要
对于每个均衡器$ v $，在同一希尔伯特空间中作用于希尔伯特空间，一对矢量$ f $和$ g $，我们将非负数$ c（v; f，g）$与\ [c（v; f，g）定义，g \ rangle |^2。 \]我们证明，当\ [c（v; f，g）\ neq 0。\ \]时，排名一驱动$ v + f \ otimes g $是可以左右的。在这里，Shift指的是通过坐标函数$ z $的乘法运算符。最后，我们检查了$ d + f \ otimes g $，其中$ d $是具有非零对角线条目的对角线运算符，$ f $和$ g $是具有非零傅立叶系数的向量。我们证明$ d + f \ otimes g $在且仅当$ d + f \ otimes g $是可逆的时才可静置。
For each isometry $V$ acting on some Hilbert space and a pair of vectors $f$ and $g$ in the same Hilbert space, we associate a nonnegative number $c(V;f,g)$ defined by \[ c(V; f,g) = (\|f\|^2 - \|V^*f\|^2) \|g\|^2 + |1 + \langle V^*f , g\rangle|^2. \] We prove that the rank-one perturbation $V + f \otimes g$ is left-invertible if and only if \[ c(V;f,g) \neq 0. \] We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function $z$. Finally, we examine $D + f \otimes g$, where $D$ is a diagonal operator with nonzero diagonal entries and $f$ and $g$ are vectors with nonzero Fourier coefficients. We prove that $D + f\otimes g$ is left-invertible if and only if $D+f\otimes g$ is invertible.
