Compression (functional Analysis)

In functional analysis, the compression of a linear operator ''T'' on a Hilbert space to a subspace ''K'' is the operator

:$P_K T \vert_K : K \rightarrow K$,

where $P_K : H \rightarrow K$ 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 $H$ and an isometry ''V'' on a subspace $W$ of $H$, define the compression of ''T'' to $W$ by

:$T_W = V^*TV : W \rightarrow W$,

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.

Functional analysis

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