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

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

The defect indices of T are the pair

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

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

${\displaystyle T=\Gamma \oplus U}$

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^[1]

${\displaystyle \displaystyle ((\mathcal {H))=H\oplus H\oplus H\oplus \cdots ,))$

the orthogonal direct sum of countably many copies of H.

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

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

Let

${\displaystyle \displaystyle ((\mathcal {K))={\mathcal {H))\oplus {\mathcal {H)).))$

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

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

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

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.^[2]

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

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

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

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

where

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

Moreover,

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

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

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

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

${\displaystyle \displaystyle {U(g)f(x)=f(g^{-1}x).))$

Moreover, H can be identified with a closed subspace of ${\displaystyle {\mathcal {H))}$ using the isometric embedding sending v in H to f_v with

${\displaystyle f_{v}(g)=\delta _{g,1}v.\,}$

If P is the projection of ${\displaystyle {\mathcal {H))}$ onto H, then

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

using the above identification.

When G is a separable topological group, Φ is continuous in the strong (or weak) operator topology if and only if U is.

In this case functions supported on a countable dense subgroup of G are dense in ${\displaystyle {\mathcal {H))}$, so that ${\displaystyle {\mathcal {H))}$ is separable.

When G = Z any contraction operator T defines such a function Φ through

${\displaystyle \displaystyle \Phi (0)=I,\,\,\,\Phi (n)=T^{n},\,\,\,\Phi (-n)=(T^{*})^{n},}$

for n > 0. The above construction then yields a minimal unitary dilation.

The same method can be applied to prove a second dilation theorem of Sz._Nagy for a one-parameter strongly continuous contraction semigroup T(t) (t ≥ 0) on a Hilbert space H. Cooper (1947) had previously proved the result for one-parameter semigroups of isometries,^[3]

The theorem states that there is a larger Hilbert space K containing H and a unitary representation U(t) of R such that

${\displaystyle \displaystyle {T(t)=PU(t)P))$

and the translates U(t)H generate K.

In fact T(t) defines a continuous operator-valued positove-definite function Φ on R through

${\displaystyle \displaystyle {\Phi (0)=I,\,\,\,\Phi (t)=T(t),\,\,\,\Phi (-t)=T(t)^{*},))$

for t > 0. Φ is positive-definite on cyclic subgroups of R, by the argument for Z, and hence on R itself by continuity.

The previous construction yields a minimal unitary representation U(t) and projection P.

The Hille-Yosida theorem assigns a closed unbounded operator A to every contractive one-parameter semigroup T'(t) through

${\displaystyle \displaystyle {A\xi =\lim _{t\downarrow 0}{1 \over t}(T(t)-I)\xi ,))$

where the domain on A consists of all ξ for which this limit exists.

A is called the generator of the semigroup and satisfies

${\displaystyle \displaystyle {-\Re (A\xi ,\xi )\geq 0))$

on its domain. When A is a self-adjoint operator

${\displaystyle \displaystyle {T(t)=e^{At},))$

in the sense of the spectral theorem and this notation is used more generally in semigroup theory.

The cogenerator of the semigroup is the contraction defined by

${\displaystyle \displaystyle {T=(A+I)(A-I)^{-1}.))$

A can be recovered from T using the formula

${\displaystyle \displaystyle {A=(T+I)(T-I)^{-1}.))$

In particular a dilation of T on K ⊃ H immediately gives a dilation of the semigroup.^[4]

Functional calculus

Let T be totally non-unitary contraction on H. Then the minimal unitary dilation U of T on K ⊃ H is unitarily equivalent to a direct sum of copies the bilateral shift operator, i.e. multiplication by z on L²(S¹).^[5]

If P is the orthogonal projection onto H then for f in L^∞ = L^∞(S¹) it follows that the operator f(T) can be defined by

${\displaystyle \displaystyle {f(T)\xi =Pf(U)\xi .))$

Let H^∞ be the space of bounded holomorphic functions on the unit disk D. Any such function has boundary values in L^∞ and is uniquely determined by these, so that there is an embedding H^∞ ⊂ L^∞.

For f in H^∞, f(T) can be defined without reference to the unitary dilation.

In fact if

${\displaystyle \displaystyle {f(z)=\sum _{n\geq 0}a_{n}z^{n))}$

for |z| < 1, then for r < 1

${\displaystyle \displaystyle {f_{r}(z))=\sum _{n\geq 0}r^{n}a_{n}z^{n))}$

is holomorphic on |z| < 1/r.

In that case f_r(T) is defined by the holomorphic functional calculus and f (T ) can be defined by

${\displaystyle \displaystyle {f(T)\xi =\lim _{r\rightarrow 1}f_{r}(T)\xi .))$

The map sending f to f(T) defines an algebra homomorphism of H^∞ into bounded operators on H. Moreover, if

${\displaystyle \displaystyle {f^{\sim }(z)=\sum _{n\geq 0}a_{n}{\overline {z))^{n},))$

then

${\displaystyle \displaystyle {f^{\sim }(T)=f(T^{*})^{*}.))$

This map has the following continuity property: if a uniformly bounded sequence f_n tends almost everywhere to f, then f_n(T) tends to f(T) in the strong operator topology.

For t ≥ 0, let e_t be the inner function

${\displaystyle \displaystyle {e_{t}(z)=\exp t{z+1 \over z-1}.))$

If T is the cogenerator of a one-parameter semigroup of completely non-unitary contractions T(t), then

${\displaystyle \displaystyle {T(t)=e_{t}(T)))$

and

${\displaystyle \displaystyle {T={1 \over 2}I-{1 \over 2}\int _{0}^{\infty }e^{-t}T(t)\,dt.))$

C₀ contractions

A completely non-unitary contraction T is said to belong to the class C₀ if and only if f(T) = 0 for some non-zero f in H^∞. In this case the set of such f forms an ideal in H^∞. It has the form φ ⋅ H^∞ where g is an inner function, i.e. such that |φ| = 1 on S¹: φ is uniquely determined up to multiplication by a complex number of modulus 1 and is called the minimal function of T. It has properties analogous to the minimal polynomial of a matrix.

The minimal function φ admits a canonical factorization

${\displaystyle \displaystyle {\varphi (z)=cB(z)e^{-P(z)},))$

where |c|=1, B(z) is a Blaschke product

${\displaystyle \displaystyle {B(z)=\prod \left[{|\lambda _{i}| \over \lambda _{i)){\lambda _{i}-z \over 1-{\overline {\lambda ))_{i))\right]^{m_{i)),))$

with

${\displaystyle \displaystyle {\sum m_{i}(1-|\lambda _{i}|)<\infty ,))$

and P(z) is holomorphic with non-negative real part in D. By the Herglotz representation theorem,

${\displaystyle \displaystyle {P(z)=\int _{0}^{2\pi }{1+e^{-i\theta }z \over 1-e^{-i\theta }z}\,d\mu (\theta )))$

for some non-negative finite measure μ on the circle: in this case, if non-zero, μ must be singular with respect to Lebesgue measure. In the above decomposition of φ, either of the two factors can be absent.

The minimal function φ determines the spectrum of T. Within the unit disk, the spectral values are the zeros of φ. There are at most countably many such λ_i, all eigenvalues of T, the zeros of B(z). A point of the unit circle does not lie in the spectrum of T if and only if φ has a holomorphic continuation to a neighborhood of that point.

φ reduces to a Blaschke product exactly when H equals the closure of the direct sum (not necessarily orthogonal) of the generalized eigenspaces^[6]

${\displaystyle \displaystyle {H_{i}=\{\xi :(T-\lambda _{i}I)^{m_{i))\xi =0\}.))$

Quasi-similarity

Two contractions T₁ and T₂ are said to be quasi-similar when there are bounded operators A, B with trivial kernel and dense range such that

${\displaystyle \displaystyle {AT_{1}=T_{2}A,\,\,\,BT_{2}=T_{1}B.))$

The following properties of a contraction T are preserved under quasi-similarity:

• being unitary
• being completely non-unitary
• being in the class C₀
• being multiplicity free, i.e. having a commutative commutant

Two quasi-similar C₀ contractions have the same minimal function and hence the same spectrum.

The classification theorem for C₀ contractions states that two multiplicity free C₀ contractions are quasi-similar if and only if they have the same minimal function (up to a scalar multiple).^[7]

A model for multiplicity free C₀ contractions with minimal function φ is given by taking

${\displaystyle \displaystyle {H=H^{2}\ominus \varphi H^{2},))$

where H² is the Hardy space of the circle and letting T be multiplication by z.^[8]

Such operators are called Jordan blocks and denoted S(φ).

As a generalization of Beurling's theorem, the commutant of such an operator consists exactly of operators ψ(T) with ψ in H^≈, i.e. multiplication operators on H² corresponding to functions in H^≈.

A C₀ contraction operator T is multiplicity free if and only if it is quasi-similar to a Jordan block (necessarily corresponding the one corresponding to its minimal function).

Examples.

• If a contraction T if quasi-similar to an operator S with

${\displaystyle \displaystyle {Se_{i}=\lambda _{i}e_{i))}$

with the λ_i's distinct, of modulus less than 1, such that

${\displaystyle \displaystyle {\sum (1-|\lambda _{i}|)<1))$

and (e_i) is an orthonormal basis, then S, and hence T, is C₀ and multiplicity free. Hence H is the closure of direct sum of the λ_i-eigenspaces of T, each having multiplicity one. This can also be seen directly using the definition of quasi-similarity.

• The results above can be applied equally well to one-parameter semigroups, since, from the functional calculus, two semigroups are quasi-similar if and only if their cogenerators are quasi-similar.^[9]

Classification theorem for C₀ contractions: Every C₀ contraction is canonically quasi-similar to a direct sum of Jordan blocks.

In fact every C₀ contraction is quasi-similar to a unique operator of the form

${\displaystyle \displaystyle {S=S(\varphi _{1})\oplus S(\varphi _{1}\varphi _{2})\oplus S(\varphi _{1}\varphi _{2}\varphi _{3})\oplus \cdots ))$

where the φ_n are uniquely determined inner functions, with φ₁ the minimal function of S and hence T.^[10]