Compact operator on Hilbert space

In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By 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 to prove e.g. the spectral theorem are different, involving operator valued measures on the spectrum.

Some results for compact operators on Hilbert space will be discussed, starting with general properties before considering subclasses of compact operators.

Let H be a Hilbert space, L(H) be the bounded operators on H. T ∈ L(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: X → Y 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 topology). (See (Zhu 2007, Theorem 1.14, p.11), and note in this reference that the uniform boundedness will apply in the situation where F ⊆ X satisfies (∀φ ∈ Hom(X, K)) sup{x**(φ) = φ(x):x} < ∞, where K is the underlying field. The uniform boundedness principle applies since Hom(X, K) with the norm topology will be a Banach space, and the maps x**:Hom(X,K) → K are continuous homomorphisms with respect to this topology.)

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