Unitary Dilations

Abstract

Suppose that H is a subspace of a Hilbert space K, and let P be the (orthogonal) projection from K onto H. Each operator B on K induces in a natural way an operator A on H defined for each f in H by

$$Af\, = \,PBf.$$

The relation between A and B can also be expressed by

$$AP\, = \,PBP.$$

Under these conditions the operator A is called the compression of B to H and B is called a dilation of A to K. This geometric definition of compression and dilation is to be contrasted with the customary concepts of restriction and extension: if it happens that H is invariant under B, then it is not necessary to project Bf back into H (it is already there), and, in that case, A is the restriction of B to H and B is an extension of A to K. Restriction-extension is a special case of compression-dilation, the special case in which the operator on the larger space leaves the smaller space invariant.