Compression (functional analysis)

In functional analysis, the term "compression" refers to several related concepts, all involving restricting the domain and range of an operator. The precise meaning depends on the context. Two common types of compression are described below.

1. Compression of an Operator to a Subspace:

Let H be a Hilbert space and P be an orthogonal projection onto a closed subspace M of H. If T is a bounded linear operator on H, then the compression of T to M is the operator T_M defined on M by:

T_M(x) = P(T(x)) for all x in M.

In other words, we apply the operator T to a vector in M, and then project the result back onto M. T_M can be seen as a restriction of T "to M", followed by projection onto M to ensure the image is contained within M.

Key properties and uses of the compression T_M:

• T_M is a bounded linear operator on M. Its norm is bounded by the norm of T.
• The compression provides a way to study the behavior of T on a smaller, potentially more manageable subspace.
• This concept is useful in studying spectral properties of operators. For example, one may deduce properties of the spectrum of T from the spectrum of T_M.
• The compression can simplify calculations and approximations of operators.
• In operator theory, studying the compression to invariant subspaces is particularly important.

2. Canonical Compression:

This relates to representing an operator as a sub-operator of a larger operator on a larger Hilbert space. Let T: H → K be a bounded linear operator between Hilbert spaces H and K. A canonical compression of T is an operator S: H' → K', where H' contains H as a subspace and K' contains K as a subspace, and further, the restriction of S to H, followed by the projection onto K, yields T.

Specifically, we can embed H into H' by considering an isometry i: H → H', and similarly embed K into K' with an isometry j: K → K'. Then, S is a canonical compression of T if for all x in H:

j(T(x)) = P_K'(S(i(x))),

where P_K' is the orthogonal projection from K' onto j(K), and we can identify j(K) with K.

This idea is often used in operator dilation theory, where one seeks to represent an operator T as a part of a "larger" operator S with better properties. The properties of S can then be used to deduce information about T.