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As such, results from matrix theory can sometimes be extended to compact operators using similar arguments.
Compact operators on a Banach space are always completely continuous.
The set of all compact operators on E is a closed ideal in this algebra.
This is essential in developing the spectral properties of compact operators.
An equivalent definition of compact operators on a Hilbert space may be given as follows.
We list some general properties of compact operators.
The notion of singular values can be extended from matrices to compact operators.
The compact operators form an important class of bounded operators.
A corresponding result holds for normal compact operators on Hilbert spaces.
Invariance by perturbation is true for larger classes than the class of compact operators.
Any compact operator is strictly singular, but not vice-versa.
Compact operators form an ideal in the ring of bounded operators.
Since is a compact operator, is infinitely close to and therefore one has also .
The result also holds for compact operators.
However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators.
By the norm-closedness of the ideal of compact operators, the converse is also true.
The spectral theory of compact operators was first developed by F. Riesz.
In this theory, the kernel is understood to be a compact operator acting on a Banach space of functions.
From Fredholm theory, smooth integral kernels are compact operators.
"the extension to polynomially compact operators was obtained by Bernstein and Robinson (1966).
An important subclass of compact operators is the trace-class or nuclear operators.
The second decomposition generalizes more easily for general compact operators on Banach spaces.
In more precise terms, the Fredholm alternative only applies when K is a compact operator.
This includes the case of polynomially compact operators because an operator commutes with any polynomial in itself.
Observe that , the algebra of compact operators.
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