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 {\displaystyle P_{K}T\vert _{K}:K\rightarrow K} , where P K : H → K {\displaystyle 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→K sending k to Tk. More generally, for a linear operator T on a Hilbert space H {\displaystyle H} and an isometry V on a subspace W {\displaystyle W} of H {\displaystyle H} , define the compression of T to W {\displaystyle W} by
T W = V ∗ T V : W → W {\displaystyle T_{W}=V^{*}TV:W\rightarrow W} , where V ∗ {\displaystyle V^{*}} is the adjoint of V. If T is a self-adjoint operator, then the compression T W {\displaystyle T_{W}} is also self-adjoint. When V is replaced by the inclusion map I : W → H {\displaystyle I:W\to H} , V ∗ = I ∗ = P K : H → W {\displaystyle V^{*}=I^{*}=P_{K}:H\to W} , and we acquire the special definition above.
See also Dilation (operator theory)
References P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.