Editing Compact operator (section)

==Compact operator on Hilbert spaces==
{{main|Compact operator on Hilbert space}}
For Hilbert spaces, another equivalent definition of compact operators is given as follows.

An operator <math>T</math> on an infinite-dimensional [[Hilbert space]] <math>(\mathcal{H}, \langle \cdot, \cdot \rangle)</math>,

:<math>T\colon\mathcal{H} \to \mathcal{H}</math>,

is said to be ''compact'' if it can be written in the form

:<math>T = \sum_{n=1}^\infty \lambda_n \langle f_n, \cdot \rangle g_n</math>,

where <math>\{f_1,f_2,\ldots\}</math> and <math>\{g_1,g_2,\ldots\}</math> are orthonormal sets (not necessarily complete), and <math>\lambda_1,\lambda_2,\ldots</math> is a sequence of positive numbers with limit zero, called the [[singular value decomposition#Bounded operators on Hilbert spaces|singular value]]s of the operator, and the series on the right hand side converges in the operator norm. The singular values can [[limit point|accumulate]] only at zero. If the sequence becomes stationary at zero, that is <math>\lambda_{N+k}=0</math> for some <math>N \in \N</math> and every <math>k = 1,2,\dots</math>, then the operator has finite rank, ''i.e.'', a finite-dimensional range, and can be written as

:<math>T = \sum_{n=1}^N \lambda_n \langle f_n, \cdot \rangle g_n</math>.

An important subclass of compact operators is the [[trace class|trace-class]] or [[nuclear operator]]s, i.e., such that <math>\operatorname{Tr}(|T|)<\infty</math>. While all trace-class operators are compact operators, the converse is not necessarily true. For example <math display="inline">\lambda_n = \frac{1}{n}</math> tends to zero for <math>n \to \infty</math> while <math display="inline">\sum_{n=1}^{\infty} |\lambda_n| = \infty</math>.