On the invertibility of general Wiener-Hopf operators

Abstract:

Let $\mathfrak {H}$ be a separable Hilbert space, $\mathfrak {B}$ the set of bounded linear operators on $\mathfrak {H}$, and $P$ an orthogonal projection on $\mathfrak {H}$. Denote the range of $P$ by $R(P)$. Let $A$ belong to $\mathfrak {B}$. The general Wiener-Hopf operator associated with $A$ and $P$ is defined by ${T_P}(A) = PA|R(P)$, the vertical bar denoting restriction. Let $Q = I - P$. The purpose of this paper is to disprove the general conjecture that if $A$ is an invertible element of $\mathfrak {B}$, then the invertibility of ${T_P}(A)$ implies the invertibility of ${T_Q}(A)$. We also disprove the conjecture in an interesting special case.