Compact Operator - Important Properties

Important Properties

In the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with the operator norm, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X), is the identity operator on X.

• K(X, Y) is a closed subspace of B(X, Y): Let T_n, n ∈ N, be a sequence of compact operators from one Banach space to the other, and suppose that T_n converges to T with respect to the operator norm. Then T is also compact.

•   In particular, K(X) forms a two-sided operator ideal in B(X).

• is compact if and only if X has finite dimension.

• For any T ∈ K(X),   is a Fredholm operator of index 0. In particular,   is closed. This is essential in developing the spectral properties of compact operators. One can notice the similarity between this property and the fact that, if M and N are subspaces of a Banach space where M is closed and N is finite dimensional, then M + N is also closed.

• Any compact operator is strictly singular, but not vice-versa.