Compression (functional analysis)

In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator


 * $$P_K T \vert_K : K \rightarrow K $$,

where $$P_K : H \rightarrow K$$ 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&rarr;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


 * $$T_W = V^*TV : W \rightarrow W$$,

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 $$I: W \to H$$, $$V^* = I^*=P_K : H \to W$$, and we acquire the special definition above.