Compression (functional analysis)
In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator
- <math>P_K T \vert_K : K \rightarrow K </math>,
where <math>P_K : H \rightarrow K</math> 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 <math>H</math> and an isometry V on a subspace <math>W</math> of <math>H</math>, define the compression of T to <math>W</math> by
- <math>T_W = V^*TV : W \rightarrow W</math>,
where <math>V^*</math> is the adjoint of V. If T is a self-adjoint operator, then the compression <math>T_W</math> is also self-adjoint. When V is replaced by the inclusion map <math>I: W \to H</math>, <math>V^* = I^*=P_K : H \to W</math>, and we acquire the special definition above.
See alsoEdit
ReferencesEdit
- P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.