On the Solvability of Some Operator Equations

Abstract

We consider the Volterra integration operator $V$, $Vf( x)=\int_{0}^{x} f( t)\mkern1mu dt$, on the following subspace of the Wiener algebra $W[ 0,1]$:\[W^{( 1) }[ 0,1] :=\{ f\in W[ 0,1] : f^{\prime}\!\in W[ 0,1] \} .\]We investigate solvability of the operator equations $VA=\lambda AV$ and $VA=\lambda A^{2}V$, where $\lambda\in\mathbb{C}$ is a complex number. Our proof is based onthe Duhamel product of functions defined by\[( f \circledast g) ( x) :=\frac{d}{dx}\int_{0}^{x}f( x-t) g( t) \, dt.\]