秩1扰动的左可逆性
Left-invertibility of rank-one perturbations
摘要：
对于作用在某个Hilbert空间上的等距$V$和同一Hilbert空间中的一对向量$f$和$g$，我们关联一个由\[ c(V; f,g) = (\|f\|^2 - \|V^*f\|^2) \|g\|^2 + |1 + \langle V^*f , g\rangle|^2. \]定义的非负数$c(V;f,g)$。我们证明了秩一扰动$V + f \otimes g$左可逆的当且仅当\[ c(V;f,g) \neq 0. \]。这里，移位是指与坐标函数PDMATHAMT相乘的运算符。最后，我们研究了$D + f \otimes g$，其中$D$是具有非零对角项的对角算子，$f$和$g$是具有非零傅立叶系数的向量。证明了$D + f\otimes g$是左可逆的当且仅当$D+f\otimes g$是可逆的.
英文摘要：
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.
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