Contraction (operator theory)

In operator theory, a discipline within mathematics, 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.

Contractions on a Hilbert space

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 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 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{\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 generalisations.^[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{\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\ge 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

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

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

f_v(g)=\delta_{g,1} v. \,

If P is the projection of ${\mathcal {H}}$ 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 ${\mathcal {H}}$ , so that ${\mathcal {H}}$ 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. 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{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 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{-\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.^[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{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{f(z)=\sum_{n\ge 0} a_n z^n}

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

\displaystyle{f_r(z))=\sum_{n\ge 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{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 {f^{\sim }(z)=\sum _{{n\geq 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_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{\varphi(z) = c B(z) e^{-P(z)},}

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

\displaystyle{B(z)=\prod \left[{|\lambda_i|\over \lambda_i} {\lambda_i -z \over 1-\overline{\lambda}_i }\right]^{m_i},}

with

\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{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 neighbourhood 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{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{AT_1=T_2A,\,\,\, BT_2=T_1B.}

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

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{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 multiplcity 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=\lambda_i e_i}

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

\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{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]

Notes

References

Bercovici, H. (1988), Operator theory and arithmetic in H^∞, Mathematical Surveys and Monographs, 26, American Mathematical Society, ISBN 0-8218-1528-8
Cooper, J. L. B. (1947), "One-parameter semigroups of isometric operators in Hilbert space", Ann. of Math., 48: 827–842, doi:10.2307/1969382
Gamelin, T. W. (1969), Uniform algebras, Prentice-Hall
Hoffman, K. (1962), Banach spaces of analytic functions, Prentice-Hall
Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Kérchy, L. (2010), Harmonic analysis of operators on Hilbert space, Universitext (Second ed.), Springer, ISBN 978-1-4419-6093-1
Riesz, F.; Sz.-Nagy, B. (1995), Functional analysis. Reprint of the 1955 original, Dover Books on Advanced Mathematics, Dover, pp. 466–472, ISBN 0-486-66289-6

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