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. This notion is a special case of the concept of a contraction mapping, but 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.

Contractions on a Hilbert space

"Operator angle" redirects here. For other uses, see Camera angle.

If T is a contraction acting on a Hilbert space ${\mathcal {H}}$, the following basic objects associated with T can be defined.

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 closure of the ranges Ran(D_T) and Ran(D_T*) respectively. The positive operator D_T induces an inner product on ${\mathcal {H}}$. The inner product space can be identified naturally with Ran(D_T). A similar statement holds for ${\mathcal {D}}_{T*}$.

The defect indices of T are the pair

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

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 non-zero 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.

Dilation theorem for contractions

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.

In fact define

$\displaystyle {{\mathcal {H}}=H\oplus H\oplus H\oplus \cdots ,}$

the orthogonal direct sum of countably many copies of H.

Let V be the isometry on ${\mathcal {H}}$ defined by

$\displaystyle {V(\xi _{1},\xi _{2},\xi _{3},\dots )=(T\xi _{1},{\sqrt {I-T^{*}T}}\xi _{1},\xi _{2},\xi _{3},\dots ).}$

Let

$\displaystyle {{\mathcal {K}}={\mathcal {H}}\oplus {\mathcal {H}}.}$

Define a unitary W on ${\mathcal {K}}$ by

$\displaystyle {W(x,y)=(Vx+(I-VV^{*})y,-V^{*}y).}$

W is then a unitary dilation of T with H considered as the first component of ${\mathcal {H}}\subset {\mathcal {K}}$.

The minimal dilation U is obtained by taking the restriction of W to the closed subspace generated by powers of W applied to H.

Dilation theorem for contraction semigroups

There is an alternative proof of Sz.-Nagy's dilation theorem, which allows significant generalization.

Let G be a group, U(g) a unitary representation of G on a Hilbert space K and P an orthogonal projection onto a closed subspace H = PK of K.

The operator-valued function

$\displaystyle {\Phi (g)=PU(g)P,}$

with values in operators on K satisfies the positive-definiteness condition

$\sum \lambda _{i}{\overline {\lambda _{j}}}\Phi (g_{j}^{-1}g_{i})=PT^{*}TP\geq 0,$

where

$\displaystyle {T=\sum \lambda _{i}U(g_{i}).}$

Moreover,

$\displaystyle {\Phi (1)=P.}$

Conversely, every operator-valued positive-definite function arises in this way. Recall that every (continuous) scalar-valued positive-definite function on a topological group induces an inner product and group representation φ(g) = 〈U_g v, v〉 where U_g is a (strongly continuous) unitary representation (see Bochner's theorem). Replacing v, a rank-1 projection, by a general projection gives the operator-valued statement. In fact the construction is identical; this is sketched below.

Let ${\mathcal {H}}$ be the space of functions on G of finite support with values in H with inner product

$\displaystyle {(f_{1},f_{2})=\sum _{g,h}(\Phi (h^{-1}g)f_{1}(g),f_{2}(h)).}$

G acts unitarily on ${\mathcal {H}}$ by

$U(g)f(x)=f({}^{}$