Compression (functional analysis)

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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 identity function I: W -> H, V^* = I^*=P_K : H -> W, and we acquire the special definition above.

See also[edit]

References[edit]

  • P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.