Contraction (operator theory) explained

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

l{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

l{D} _{T} and l{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 l{H} . The inner product space can be identified naturally with Ran(D_T). A similar statement holds for l{D} _{T*} .

The defect indices of T are the pair

(\dim l{D} _{T,} \dim l{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 ⊕ 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{ l{H}=H ⊕ H ⊕ H ⊕ … ,}

the orthogonal direct sum of countably many copies of H.

Let V be the isometry on

l H defined by\displaystyle{V(\xi _{1,\xi} _{2,\xi} _{3,...)=(T\xi} _{1,} \sqrt{I - T ^{*T}\xi} _{1 ,\xi} _{2,\xi} _{3,...).}}

Let

\displaystyle{ l{K}= l{H} ⊕ l{H}.}

Define a unitary W on

l 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

l{H}\subset l{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 λ _{i\overline{λ} _{j}} \Phi(g ^{-1} _{j} g _{i)} = PT ^{*TP\ge} 0 ,

where

\displaystyle{T = \sum λ _{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

l 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

l H by\displaystyle{U(g)f(x) = f(g ^{-1} x).}

Moreover, H can be identified with a closed subspace of

l H using the isometric embeddingsending v in H to f_v withf _{v(g)=\delta} _{g,1} v.

If P is the projection of

l H onto H, then\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

l H , so that l H is separable.

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

\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. had previously proved the result for one-parameter semigroups of isometries,

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

\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{\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{A\xi = \lim _{t\downarrow} { 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{ - \Re (A\xi,\xi)\ge 0 }

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

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

A can be recovered from T using the formula

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

In particular a dilation of T on K ⊃ H immediately gives a dilation of the semigroup.

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).

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

\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 definedwithout reference to the unitary dilation.

In fact if

\displaystyle{f(z) = \sum _{n\ge} a _{n} z ^{n}}

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

\displaystyle{f _{r(z))=\sum} _{n\ge} 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{f(T)\xi = \lim _{r →} 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{f ^{\sim(z) = \sum} _{n\ge 0} a _{n} \overline{z} ^{n,}}

then

\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{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{T(t) = e _{t(T)}}

and

\displaystyle{T = { 1 \over 2 }I - { 1 \over 2 }\int ^{inf ty} _{0} 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-zerof 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{\varphi(z) = c B(z) e ^{-P(z)} ,}

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

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

with

\displaystyle{\sum m _{i(1-|λ} _{i|)} < inf ty,} and P(z) is holomorphic with non-negative real part in D. By the Herglotz representation theorem,\displaystyle{P(z) = \int ^{2 \pi} _{0} { 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

\displaystyle{H _{i=\{\xi:(T-λ} _{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{AT _{1=T} _{2A,} BT _{2=T} _{1B.}}

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).

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

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

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

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{Se _{i=λ} _{i} e _{i}}

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

\displaystyle{\sum ( 1- | λ _{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.

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{S = S(\varphi _{1) ⊕} S(\varphi _{1\varphi} _{2) ⊕} S(\varphi _{1\varphi} _{2\varphi} _{3)} ⊕ … }

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

See also

• • • • Hille-Yosida theorem for contraction semigroups