Approximation of approximation numbers by truncation

Abstract

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk≥1, the approximation numberss k(An) converge to the corresponding approximation numbers k(A) asn→∞. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA n asn goes to infinity.