Inverse-closedness of subalgebras of integral operators with almost periodic kernels (Q2299286)

Let \(\mathbb{E}\) be a complex Banach space and \(\mathcal{B}(\mathbb{E})\) be the Banach algebra of all bounded linear operators on \(\mathbb{E}\). The class of kernels \(n:\mathbb{R}^d\times\mathbb{R}^d\to\mathcal{B}(\mathbb{E})\) represented in the form \(n(x,y)=\sum_{k=1}^\infty e^{i\langle \omega_k,x\rangle}n_k(y)\), where \(\omega_k\in\mathbb{R}^d\) and \(n_k\in L^1(\mathbb{R}^d,\mathcal{B}(\mathbb{E}))\) with \(\sum_{k=1}^\infty\|n_k\|_{L^1}<\infty\), is denoted by \(\mathbf{CN}_{1,APW}(\mathbb{R}^d,\mathbb{E})\). For \(1\le p\le\infty\), the set of all operators \(N\in\mathcal{B}(L^p)\) of the form \(\alpha I+\int_{\mathbb{R}^d}n(x,x-y)u(y)\,dy\) with \(\alpha\in\mathbb{C}\) and \(n\in \mathbf{CN}_{1,APW}(\mathbb{R}^d,\mathbb{E})\) is denoted by \(\widetilde{\mathbf{CN}}_{1,APW}(L^p)\). The main result of the paper says that \(\widetilde{\mathbf{CN}}_{1,APW}(L^p)\) is an inverse closed subalgebra of \(\mathcal{B}(L^p)\).