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.