title: Compression (functional analysis)
text: # 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.
  
  