Contraction (operator theory)
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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 
. 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 
and 
are the ranges Ran(DT) and Ran(DT*) respectively. The positive operator DT induces an inner product on . The space can be identified naturally with , with the induced inner product. The same can be said for .
The defect indices of T are the pair
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
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.
