On the algebra of pair integral operators with homogeneous kernels (Q1886122)

The author investigates the smallest closed subalgebra \(\mathcal{A}\) of \({\mathcal L} (L^p({\mathbb R}^n))\), \(p\in [1,\infty]\), that contains all operators of the form \(A=\lambda I-K_1 P-K_2 Q\), where \(\lambda \in {\mathbb C}\), \(P\) and \(Q\) are the operators of multiplication by the characteristic functions of \(| x| \leq 1\) and \(| x| >1\), respectively, and \(K_j\), \(j=1,2\), is an integral operator whose kernel \(k_j(x,y)\) satisfies the following three conditions: (C1) \(k_j\) is homogeneneous of degree \(-n\), i.e., \(k_j (\alpha x, \alpha y)=\alpha^{- n} k_ j (x, y)\; \forall \alpha>0\); (C2) \(k_j\) is invariant with respect to the rotation group \(SO(n)\) of \({\mathbb R}^ n\), i.e., \(k_j (\omega(x),\omega(y)) = k_j (x, y)\) \(\forall\omega \in SO(n)\); (C3) \(k_j\) is summable, i.e., \(k_j= \int_{{\mathbb R}^n} | k_j(e_1,y)| | y| ^{- n/p} dy < \infty\), where \(e_1=(1,0,\ldots,0)\). Let \(\mathcal T\) be the set of all compact operators in \(\mathcal{A}\). It is shown that the quotient algebra \(\mathcal{A}/\mathcal{T}\) is commutative, the maximal ideal space is described and the Gelfand transform of the elements in \(\mathcal{A}/\mathcal{T}\) is identified. This permits to state and prove the main result of the paper (Theorem 5), a Fredholm criterion and an index formula for the operators in \(\mathcal{A}\). The matrix case is also considered at the end of the paper.