View source for Spectral theory of compact operators

''Riesz theory of compact operators''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304701.png" /> be a complex [[Banach space|Banach space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304702.png" /> a [[Compact operator|compact operator]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304703.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304704.png" />, the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304705.png" />, is countable and has no cluster points except, possibly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304706.png" />. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304707.png" /> is an eigenvalue, and a pole of the resolvent function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304708.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s1304709.png" /> be the order of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047010.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047012.png" /> is closed, and this range is constant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047013.png" />. The null space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047014.png" /> is finite dimensional and constant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047015.png" />. The spectral projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047016.png" /> (the Riesz projector, see [[Riesz decomposition theorem|Riesz decomposition theorem]]) has non-zero finite-dimensional range, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047017.png" />, and its null space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047018.png" />. Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047019.png" />.

The respective integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047021.png" /> are called the index and the algebraic multiplicity of the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130470/s13047022.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.R. Dowson,   "Spectral theory of linear operators" , Acad. Press  (1978)  pp. 45ff.</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators I: General theory" , Interscience  (1964)  pp. Sect. VII.4</TD></TR></table>