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. General, let V be isometry on Hilbert space W, subspace of Hilbert space H (T on H). We define compression T_W of a linear operator T on a Hilbert space H to a subspace W as linear operator V ^* TV. If T is hermitian operator, then compression is hermitian too. When replace V with identity I:W − > H(that implies I ^* = P_K:H − > W), we encounter special definition.

See also

• Dilation

References

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

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