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. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias. 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. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias. If T is a contraction acting on a Hilbert space H {displaystyle {mathcal {H}}} , the following basic objects associated with T can be defined. 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 D T {displaystyle {mathcal {D}}_{T}} and D T ∗ {displaystyle {mathcal {D}}_{T*}} are the ranges Ran(DT) and Ran(DT*) respectively. The positive operator DT induces an inner product on H {displaystyle {mathcal {H}}} . The inner product space can be identified naturally with Ran(DT). A similar statement holds for D T ∗ {displaystyle {mathcal {D}}_{T*}} . 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. Sz.-Nagy's dilation theorem, proved in 1953, states that for any contraction T on a Hilbert space H, there is a unitary operator U on a larger Hilbert space K ⊇ H such that if P is the orthogonal projection of K onto H then Tn = P Un P for all n > 0. The operator U is called a dilation of T and is uniquely determined if U is minimal, i.e. K is the smallest closed subspace invariant under U and U* containing H.