Approximation of approximation numbers by truncation
- Cite this article as:
- Böttcher, A., Chithra, A.V. & Namboodiri, M.N.N. Integr equ oper theory (2001) 39: 387. doi:10.1007/BF01203320
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LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letAn 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 numberssk(An) converge to the corresponding approximation numbersk(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 ofAn 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 ofAn asn goes to infinity.