In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator where
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
is the adjoint of V. If T is a self-adjoint operator, then the compression
When V is replaced by the inclusion map
, and we acquire the special definition above.