Compression (functional analysis) explained

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 → K ,

where

P _{K} : H → 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 and an isometry V on a subspace W of H , define the compression of T to W byT _{W} = V ^{*TV} : W → 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.

See also

• Dilation

References

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