Contraction (operator theory)

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.

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 D_T = (1 − T*T)^½ and D_T* = (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(D_T) and Ran(D_T*) respectively. The positive operator D_T 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.