Convolution operators on cuspidal functions
Armand Borel · Cambridge University Press eBooks · 1997
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.