Sz.-Nagy's dilation theorem

Sz.-Nagy's dilation theorem The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction T on a Hilbert space H has a unitary dilation U to a Hilbert space K, containing H, with {displaystyle T^{n}=P_{H}U^{n}vert _{H},quad ngeq 0.} Moreover, such a dilation is unique (up to unitary equivalence) when one assumes K is minimal, in the sense that the linear span of ∪nUnH is dense in K. When this minimality condition holds, U is called the minimal unitary dilation of T.

Contents 1 Proof 2 Schaffer form 3 Remarks 4 References Proof For a contraction T (i.e., ( {displaystyle |T|leq 1} ), its defect operator DT is defined to be the (unique) positive square root DT = (I - T*T)½. In the special case that S is an isometry, DS* is a projector and DS=0, hence the following is an Sz. Nagy unitary dilation of S with the required polynomial functional calculus property: {displaystyle U={begin{bmatrix}S&D_{S^{*}}\D_{S}&-S^{*}end{bmatrix}}.} Returning to the general case of a contraction T, every contraction T on a Hilbert space H has an isometric dilation, again with the calculus property, on {displaystyle oplus _{ngeq 0}H} given by {displaystyle S={begin{bmatrix}T&0&0&cdots &\D_{T}&0&0&&\0&I&0&ddots \0&0&I&ddots \vdots &&ddots &ddots end{bmatrix}}.} Substituting the S thus constructed into the previous Sz.-Nagy unitary dilation for an isometry S, one obtains a unitary dilation for a contraction T: {displaystyle T^{n}=P_{H}S^{n}vert _{H}=P_{H}(Q_{H'}Uvert _{H'})^{n}vert _{H}=P_{H}U^{n}vert _{H}.} Schaffer form This section needs expansion. You can help by adding to it. (June 2008) The Schaffer form of a unitary Sz. Nagy dilation can be viewed as a beginning point for the characterization of all unitary dilations, with the required property, for a given contraction.

Remarks A generalisation of this theorem, by Berger, Foias and Lebow, shows that if X is a spectral set for T, and {displaystyle {mathcal {R}}(X)} is a Dirichlet algebra, then T has a minimal normal δX dilation, of the form above. A consequence of this is that any operator with a simply connected spectral set X has a minimal normal δX dilation.

To see that this generalises Sz.-Nagy's theorem, note that contraction operators have the unit disc D as a spectral set, and that normal operators with spectrum in the unit circle δD are unitary.

References Paulsen, V. (2003). Completely Bounded Maps and Operator Algebras. Cambridge University Press. Schaffer, J. J. (1955). "On unitary dilations of contractions". Proceedings of the American Mathematical Society. 6 (2): 322. doi:10.2307/2032368. JSTOR 2032368. hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Operator theoryTheorems in functional analysis