Fredholm Theory in Hilbert Space - A Concise Introductory Exposition

Abstract

This is a brief introduction to Fredholm theory for Hilbert space operators organized into ten sections. The classical partition of the spectrum into point, residual, and continuous spectra is reviewed in Section 1. Fredholm operators are introduced in Section 2, and Fredholm index in Section 3. The essential spectrum is considered in Section 4, the spectral picture is presented in Section 5, and Riesz points are discussed in Section 6. Weyl spectrum is the subject of Section 7 and, after bringing some basic results on ascent and descent in Section 8, Browder spectrum is investigated in Section 9. Finally, Weyl and Browder theorems close this expository paper in Section 10.