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 | 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

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* Dilation

## References

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

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