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 $P_{K}$ 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.