Compact operator on Hilbert space

Compact Operator On Hilbert Space

Compact operator on Hilbert space

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In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. In contrast, the study of general operators on infinite dimensional spaces often requires a genuinely different approach.

For example, the spectral theory of compact operators on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds for normal compact operators on Hilbert spaces. (More generally, the compactness assumption can be dropped. But, as stated above, the techniques used are less routine.)

This article will discuss a few results for compact operators on Hilbert space, starting with general properties before considering subclasses of compact operators.

Some general properties

Let H be a Hilbert space, L(H) be the bounded operators on H. TL(H) is a compact operator if the image of each bounded set under T is relatively compact. We list some general properties of compact operators.

If X and Y are Hilbert spaces (in fact X Banach and Y normed will suffice), then T:XY is compact if and only if it is continuous when viewed as a map from X with the weak topology to Y (with the norm...
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