Compact Operators

In these notes we provide an introduction to compact linear operators on Banach and Hilbert spaces. These operators behave very much like familiar finite dimensional matrices, without necessarily having finite rank. For more thorough treatments, see [RS, Y]. Definition 1 Let X and Y be Banach spaces. A linear operator C : X → Y is said to be compact if for each bounded sequence {xi}i∈IN ⊂ X , there is a subsequence of {Cxi}i∈IN that is convergent. Example 2 Let a < b and c < d. If C : [c, d]× [a, b] → C is continuous, then the integral operator (Cf)(y) = ∫ b a C(y, x)f(x) dx is compact as an operator from X = C[a, b], the space of continuous functions on [a, b] with supremum norm, to Y = C[c, d]. Problem 1 Use the Arzelà–Ascoli theorem to prove that the operator C of Example 2 is compact. Example 3 (Hilbert–Schmidt Operators) Let 〈X, μ〉 and 〈Y, ν〉 be measure spaces and let k(x, y) be a measurable function on X × Y with