Spectral Approximation and Index for Convolution Type Operators on Cones on $$L^{p}(\mathbb {R}^2)$$

Abstract

We consider an algebra of operator sequences containing, among others, the approximation sequences to convolution type operators on cones acting on \(L^{p}(\mathbb {R}^2)\), with 1 < p < ∞. To each operator sequence (An) we associate a family of operators \(W_{x}(A_{n}) \in \mathcal {L}(L^{p}(\mathbb {R}^2))\) parametrized by x in some index set. When all Wx(An) are Fredholm, the so-called approximation numbers of An have the α-splitting property with α being the sum of the kernel dimensions of Wx(An). Moreover, the sum of the indices of Wx(An) is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the \(\epsilon\)-pseudospectrum, norms of inverses and condition numbers are also obtained.