Toeplitz operators and differential equations on a half-line

Abstract:

Let $\mathcal {K}$ be a separable Hilbert space, let ${A_0},{A_1}, \cdots ,{A_n}$ denote bounded linear operators from $\mathcal {K}$ into $\mathcal {K}$, and let $\mathcal {D}$ represent the set of all functions in ${L^2}(0,\infty ;\mathcal {K})$ whose first n derivatives belong to ${L^2}(0,\infty ;\mathcal {K})$. Suppose further that the space $\mathcal {D}$ is equipped with an inner product inherited from ${L^2}(0,\infty ;\mathcal {K})$. The main result of this note states that the differential operator \[ L = {A_n}\frac {{{d^n}}}{{d{t^n}}} + {A_{n - 1}}\frac {{{d^{n - 1}}}}{{d{t^{n - 1}}}} + \cdots + {A_1}\frac {d}{{dt}} + {A_0}\] acting on $\mathcal {D}$ is continuously invertible if and only if the operator \[ P(\sigma ) = \sum {A_k^ \ast } {\sigma ^k}\quad (0 \leqq k \leqq n)\] acting on the Hilbert space $\mathcal {K}$ has a uniformly bounded inverse everywhere in the open half-plane $\Re \sigma < 0$.