9 - Convolution operators on cuspidal functions

Summary

Compact operators [61, X.5; 46, VI.5]. Let H be a Hilbert space (with a countable basis), ( , ) the scalar product on H, and ℒ(H) the algebra of bounded linear operators on H. If A ∈ ℒ(H) then A* denotes its adjoint – that is, the unique bounded linear operator such that (Ax, y) = (x, A*y) (x, y ∈ H). The operator A*A is self-adjoint and positive (i.e., (A*Ax, x) ≥ 0 for all x ∈ H) ; it has a unique positive square root, called the absolute value |A| of A.

The bounded operator A is compact if it transforms any bounded set into a relatively compact one. The eigenspaces of A corresponding to nonzero eigenvalues are finite dimensional. If A is compact, positive, and self-adjoint, then H has an orthonormal basis {e_i} consisting of eigenvectors of A: Ae_i = λ_i,e_i, with λ_i → 0. The operator A is compact if and only if |A| is. The compact operators obviously form an ideal in ℒ(H).

More generally, a continuous linear map A: H → H′ of H into a Hilbert space H′ is said to be compact if it transforms any bounded set into a relatively compact one or, equivalently, any bounded sequence into one containing a convergent subsequence.