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

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^[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 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 {\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

\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 )\geq 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\geq 0}a_{n}z^{n}}

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

\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 {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)=cB(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 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 {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_{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 {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 {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

^ Sz.-Nagy et al. 2010, pp. 10–14
^ Sz.-Nagy et al. 2010, pp. 24–28
^ Sz.-Nagy et al. 2010, pp. 28–30
^ Sz.-Nagy et al. 2010, pp. 143, 147
^ Sz.-Nagy et al. 2010, pp. 87–88
^ Sz.-Nagy et al. 2010, p. 138
^ Sz.-Nagy et al. 2010, pp. 395–440
^ Sz.-Nagy et al. 2010, p. 126
^ Bercovici 1988, p. 95
^ Bercovici 1988, pp. 35–66

References

Bercovici, H. (1988), Operator theory and arithmetic in H^∞, Mathematical Surveys and Monographs, vol. 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 (4): 827–842, doi:10.2307/1969382, JSTOR 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

[1] Sz.-Nagy et al. 2010, pp. 10–14

[2] Sz.-Nagy et al. 2010, pp. 24–28

[3] Sz.-Nagy et al. 2010, pp. 28–30

[4] Sz.-Nagy et al. 2010, pp. 143, 147

[5] Sz.-Nagy et al. 2010, pp. 87–88

[6] Sz.-Nagy et al. 2010, p. 138

[7] Sz.-Nagy et al. 2010, pp. 395–440

[8] Sz.-Nagy et al. 2010, p. 126

[9] Bercovici 1988, p. 95

[10] Bercovici 1988, pp. 35–66

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v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
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Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
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Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F