Compression (functional analysis)

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

${\displaystyle P_{K}T\vert _{K}:K\rightarrow K}$,

where ${\displaystyle P_{K}:H\rightarrow K}$ 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 ${\displaystyle H}$ and an isometry V on a subspace ${\displaystyle W}$ of ${\displaystyle H}$, define the compression of T to ${\displaystyle W}$ by

${\displaystyle T_{W}=V^{*}TV:W\rightarrow W}$,

where ${\displaystyle V^{*}}$ is the adjoint of V. If T is a self-adjoint operator, then the compression ${\displaystyle T_{W}}$ is also self-adjoint. When V is replaced by the inclusion map ${\displaystyle I:W\to H}$, ${\displaystyle V^{*}=I^{*}=P_{K}:H\to W}$, and we acquire the special definition above.