# Compression (functional analysis)

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

where 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 and an isometry *V* on a subspace of , define the **compression** of *T* to by

where is the adjoint of *V*. If *T* is a self-adjoint operator, then the compression is also self-adjoint.
When *V* is replaced by the identity function , , and we acquire the special definition above.

## See also

## References

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