Compact Operator On Hilbert Space - Some General Properties

Some General Properties

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 є 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.)

The family of compact operators is a norm-closed, two-sided, *-ideal in L(H). Consequently, a compact operator T cannot have a bounded inverse if H is infinite dimensional. If ST = TS = I, then the identity operator would be compact, a contradiction.

If a sequence of bounded operators S_n → S in the strong operator topology and T is compact, then S_nT converges to ST in norm. For example, consider the Hilbert space l2(N), with standard basis {e_n}. Let P_m be the orthogonal projection on the linear span of {e₁ ... e_m}. The sequence {P_m} converges to the identity operator I strongly but not uniformly. Define T by Te_n = (1/n)2 · e_n. T is compact, and, as claimed above, P_mT → I T = T in the uniform operator topology: for all x,

Notice each P_m is a finite-rank operator. Similar reasoning shows that if T is compact, then T is the uniform limit of some sequence of finite-rank operators.

By the norm-closedness of the ideal of compact operators, the converse is also true.

The quotient C*-algebra of L(H) modulo the compact operators is called the Calkin algebra, in which one can consider properties of an operator up to compact perturbation.