Contraction (operator Theory)
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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 In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ...
, 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is largely due to Béla Szőkefalvi-Nagy and
Ciprian Foias Ciprian Ilie Foiaș (20 July 1933 – 22 March 2020) was a Romanian-American mathematician. He was awarded the Norbert Wiener Prize in Applied Mathematics in 1995, for his contributions in operator theory. Education and career Born in Reșița ...
.


Contractions on a Hilbert space

If ''T'' is a contraction acting on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
\mathcal, the following basic objects associated with ''T'' can be defined. The defect operators of ''T'' are the operators ''DT'' = (1 − ''T*T'')½ and ''DT*'' = (1 − ''TT*'')½. The square root is the positive semidefinite one given by the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
. The defect spaces \mathcal_T and \mathcal_ are the closure of the ranges Ran(''DT'') and Ran(''DT*'') respectively. The positive operator ''DT'' induces an inner product on \mathcal. The inner product space can be identified naturally with Ran(''D''''T''). A similar statement holds for \mathcal_. The defect indices of ''T'' are the pair :(\dim\mathcal_T, \dim\mathcal_). 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 subspace In linear algebra, a reducing subspace W of a linear map T:V\to V from a Hilbert space V to itself is an invariant subspace of T whose orthogonal complement W^\perp is also an invariant subspace of T. That is, T(W) \subseteq W and T(W^\perp) \subse ...
s 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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
, 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 In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
''U'' on a larger Hilbert space ''K'' ⊇ ''H'' such that if ''P'' is the orthogonal projection of ''K'' onto ''H'' then ''T''''n'' = ''P'' ''U''''n'' ''P'' for all ''n'' > 0. The operator ''U'' is called a
dilation Dilation (or dilatation) may refer to: Physiology or medicine * Cervical dilation, the widening of the cervix in childbirth, miscarriage etc. * Coronary dilation, or coronary reflex * Dilation and curettage, the opening of the cervix and surgic ...
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 the orthogonal direct sum of countably many copies of ''H''. Let ''V'' be the isometry on \mathcal H defined by :\displaystyle Let :\displaystyle Define a unitary ''W'' on \mathcal K by :\displaystyle ''W'' is then a unitary dilation of ''T'' with ''H'' considered as the first component of \mathcal\subset \mathcal. 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 with values in operators on ''K'' satisfies the positive-definiteness condition : \sum \lambda_i\overline \Phi(g_j^g_i) = PT^*TP\ge 0, where :\displaystyle Moreover, :\displaystyle 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'') = 〈''Ug v'', ''v''〉 where ''Ug'' 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 ''G'' acts unitarily on \mathcal H by :\displaystyle 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_ v. \, If ''P'' is the projection of \mathcal H onto ''H'', then :\displaystyle using the above identification. When ''G'' is a separable topological group, Φ is continuous in the strong (or weak)
operator topology In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach space ...
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''. 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 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 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 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 on its domain. When ''A'' is a self-adjoint operator :\displaystyle in the sense of the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
and this notation is used more generally in semigroup theory. The cogenerator of the semigroup is the contraction defined by : \displaystyle ''A'' can be recovered from ''T'' using the formula :\displaystyle 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(''S''1). If ''P'' is the orthogonal projection onto ''H'' then for ''f'' in L = L(''S''1) it follows that the operator ''f''(''T'') can be defined by :\displaystyle 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 for , ''z'', < 1, then for ''r'' < 1 :\displaystyle 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 The map sending ''f'' to ''f''(''T'') defines an algebra homomorphism of H into bounded operators on ''H''. Moreover, if :\displaystyle then :\displaystyle 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 If ''T'' is the cogenerator of a one-parameter semigroup of completely non-unitary contractions ''T''(''t''), then :\displaystyle and :\displaystyle


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 In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In ...
, i.e. such that , φ, = 1 on ''S''1: 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 where , ''c'', =1, ''B''(''z'') is a Blaschke product :\displaystyle with :\displaystyle and ''P''(''z'') is holomorphic with non-negative real part in ''D''. By the Herglotz representation theorem, :\displaystyle for some non-negative finite measure μ on the circle: in this case, if non-zero, μ must be
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of ''T''. Within the unit disk, the spectral values are the zeros of φ. There are at most countably many such λi, 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


Quasi-similarity

Two contractions ''T''1 and ''T''2 are said to be quasi-similar when there are bounded operators ''A'', ''B'' with trivial kernel and dense range such that :\displaystyle 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 In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
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 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 ''H''2 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 with the λi's distinct, of modulus less than 1, such that :\displaystyle 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 λi-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 where the φ''n'' are uniquely determined inner functions, with φ''1'' the minimal function of ''S'' and hence ''T''.


See also

* * * * Hille-Yosida theorem for contraction semigroups


Notes


References

* * * * * * {{Hilbert space Operator theory