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]\displaystyle{ P_K T \vert_K : K \rightarrow K }[/math],

where [math]\displaystyle{ 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]\displaystyle{ H }[/math] and an isometry V on a subspace [math]\displaystyle{ W }[/math] of [math]\displaystyle{ H }[/math], define the compression of T to [math]\displaystyle{ W }[/math] by

[math]\displaystyle{ T_W = V^*TV : W \rightarrow W }[/math],

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

See also

• Dilation

References

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

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