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

  • P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.