The Noether property and the index of multidimensional convolutions with coefficients of fast oscillating type
Let End \(L_ 2({\mathbb{R}}^ m)\) denote the algebra of all linear bounded operators on \(L_ 2({\mathbb{R}}^ m)\). Let V be a closed subalgebra of End \(L_ 2({\mathbb{R}}^ m)\) generated by all convolution operators: \((Af)(t)=\int_{{\mathbb{R}}^ m}a(t-s)f(s)ds,\) where \(f\in L_ 2({\mathbb{R}}^ m)\), \(a\in L_ 1({\mathbb{R}}^ m)\). Let \(A_ f\) denote the multiplication operator by function \(f\in L_{\infty}({\mathbb{R}}^ m)\). By \(C_ pV\) we shall denote the closed subalgebra of End \(L_ 2({\mathbb{R}}^ m)\) generated by all operators of the type \(A_ fA\), where \(A\in V\) and f is a uniformly continuous function on \({\mathbb{R}}^ m\). Let us introduce the set \(\psi\) of all functions \(f\in L_{\infty}({\mathbb{R}}^ m)\) satisfying for any compact \(D\subset {\mathbb{R}}^ m\) the following condition: \(\lim_{t\to \infty}\int_{D}f(t+s)ds=0.\) Let the functions \(g_ j\), \(j=0,1,...,n\) be such that 1) \(g_ 0=1\); \(2)g_ j\bar g_ j=1\) \((j=1,...,n)\); 3) \(g_ j\bar g_ k\in \psi\) for all \(j\neq k\), \(0\leq j,k\leq n.\) The main result of the present paper is that for any \(\lambda\in {\mathbb{C}}\) and \(A_{jk}\in C_ pV\) \((j,k=0,1,...,n)\) the operator \(\lambda I+\sum^{n}_{j=0}\sum^{n}_{k=0}A_{g_ j}A_{jk}A_{\bar g_ k}\) is Noetherian if and only if the same is true for the matrix operator \(I^{(n+1)}+\{A_{jk}\}^ n_{j,k=0}\) and the indexes of these operators (in the case when they are Noetherian) coincide. (Here \(I^{(n+1)}\) is the identity operator in the \((n+1)\)-fold direct sum of \(L_ 2({\mathbb{R}}^ m))\).
scientific article; zbMATH DE number 4192452