Contraction (operator theory)

From Wikipedia, the free encyclopedia

In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators.

[edit] Contractions on a Hilbert space

Consider the special case where T is a contraction acting on a Hilbert space \mathcal{H}. We define some basic objects associated with T.

The defect operators of T are the operators DT = (1 − T*T)½ and DT* = (1 − TT*)½. The square root is the positive semidefinite one given by the spectral theorem. The defect spaces \mathcal{D}_T and \mathcal{D}_{T*} are the ranges Ran(DT) and Ran(DT*) respectively. The positive operator DT induces an inner product on \mathcal{H}. The space \mathcal{D}_T can be identified naturally with \mathcal{H}, with the induced inner product. The same can be said for \mathcal{D}_{T*}.

The defect indices of T are the pair

(\mbox{dim}\mathcal{D}_T, \mbox{dim}\mathcal{D}_{T^*}).

The defect operators and the defect indices are a measure of the non-unitarity of T.

A contraction T on a Hilbert space can be canonically decomposed into an orthogonal direct sum

T = \Gamma \oplus U

where U is a unitary operator and Γ is completely non-unitary in the sense that it has no reducing subspaces on which its restriction is unitary. If U = 0, T is said to be a completely non-unitary contraction. A special case of this decomposition is the Wold decomposition for an isometry, where Γ is a proper isometry.

Contractions on Hilbert spaces can be viewed as the operator analogs of cos θ and are called operator angles in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.