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 and an isometry V on a subspace of , define the compression of T to by
where is the adjoint of V. If T is a self-adjoint operator, then the compression is also self-adjoint.
When V is replaced by the identity function , , and we acquire the special definition above.
- P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.