In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator
P K T | K : K → K ,
where P K : H → K is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space. If K is an invariant subspace for T, then the compression of T to K is the restricted operator K→K sending k to Tk.
More generally, for a linear operator T on a Hilbert space H and an isometry V on a subspace W of H , define the compression of T to W by
T W = V ∗ T V : W → W ,
where V ∗ is the adjoint of V. If T is a self-adjoint operator, then the compression T W is also self-adjoint. When V is replaced by the inclusion map I : W → H , V ∗ = I ∗ = P K : H → W , and we acquire the special definition above.
