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

1. ,

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 KK sending k to Tk.

More generally, for a linear operator T and an isometry V, define the compression of T by

1. ,

is the adjoint of V. If T is a self-adjoint operator is also self-adjoint. When V is replaced by the inclusion map, and we acquire the special definition above.

See also

• Dilation (operator theory)

References

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