Compression (functional analysis)

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

PKT|K:K->K,

where PK: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 by

TW=V*TV:W->W,

where V* is the adjoint of V. If T is a self-adjoint operator, then the compression TW is also self-adjoint. When V is replaced by the inclusion map I:W->H, V*=I*=PK:H->W, and we acquire the special definition above.

## See also

* Dilation

## References

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

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