Left-invertibility of rank-one perturbations

TL;DR

This paper characterizes when rank-one perturbations of isometries and diagonal operators are left-invertible, providing explicit conditions involving a specific nonnegative number and exploring examples in Hilbert spaces of analytic functions.

Contribution

It introduces a necessary and sufficient condition for the left-invertibility of rank-one perturbations of isometries and diagonal operators, extending understanding of operator invertibility in Hilbert spaces.

Findings

Left-invertibility of $V + f ensor g$ is equivalent to $c(V;f,g) eq 0$.

For diagonal operators $D$, invertibility and left-invertibility of $D + f ensor g$ are equivalent.

Examples include shift operators on Hilbert spaces of analytic functions.

Abstract

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.