Spectral approximation and index for convolution type operators on cones on \(L^{p}(\mathbb {R}^2)\) (Q1049593)

The convolution type operator \[ C(a) : L^p({\mathbb R}^2) \rightarrow L^p({\mathbb R}^2), \qquad g \mapsto \lambda g(t) + \int_{{\mathbb R}^2} u(t - s) g(s) d s \] is considered. Fredholm sequences (i.e., the sequences which are regularizable by some special \(D\) consisting of sequences of compact type) in an algebra \({\mathcal E}\) of approximate sequences to the above convolution type operators are studied. It is shown that Fredholm sequences in \({\mathcal E}\) possess a splitting property of the singular values in \(L^p({\mathbb R}^2), 1 < p < \infty\). This result is obtained by defining approximation numbers in the context of infinite dimensional Banach spaces and using them instead of singular values.