The main study of this thesis is the holomorphic functional calculi for three classes of unbounded operators: sectorial, strip-type and half-plane. The functional calculus for sectorial operators was introduced by McIntosh as an extension of the Riesz-Dunford model for bounded operators. More recently Haase has developed an abstract framework which incorporates analogous constructions for strip-type and half-plane operators. These operators are of interest since they arise naturally as generators of C₀-(semi)groups. The theory of bounded H^∞-calculus for sectorial operators is well established and it has been found to have many applications in operator theory and parabolic evolution equations. We survey these known results, first on Hilbert space and then on general Banach space. Our main goal is to fill the gaps in the parallel theory for strip-type operators. Whilst some of this can be deduced by taking exponentials and applying known results for sectorial operators, in general this is insu_cient to obtain our desired results and so we pursue an independent approach. Starting on Hilbert space, we broaden known characterisations of the bounded H^∞-calculus for strip-type operators by introducing a notion of absolute calculus which is an analogue to the established notion for the sectorial case. Moving to general Banach space, we build on the work of Vörös, broadening his characterisation for strip-type operators in terms of weak integral estimates by introducing a new, but equivalent, notion of the bounded H^∞-calculus, which we call the m-bounded calculus. We also demonstrate that these characterisations fail for half-plane operators and instead present a weaker form of the bounded H-calculus which is more natural for these operators. This allows us to obtain new and simple proofs of well known generation theorems due to Gomilko and Shi-Feng, with extensions to polynomially bounded semigroups. The connection between the bounded H-calculus of semigroup generators and polynomial boundedness of their associated Cayley Transforms is also explored. Finally we present a series of results on sums of operators, in connection with maximal regularity. We also establish stability results for the bounded H^∞-calculus for strip-type operators by showing it is preserved under suitable bounded perturbations, which at time requires further assumptions on the underlying Banach space. This relies heavily on intermediate characterisations of the bounded H^∞-calculus due to Kalton and Weis.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O Teorema Espectral à um dos teoremas mais famosos da Analise Funcional, principalmente pelo grande nÃmero de versÃes dadas ao mesmo. Existem versÃes para operadores limitados, ilimitados, autoadjuntos, compactos, em espacos de dimensÃo finita ou infinita. A versÃo geral do teorema foi provada independentemente por Stone e Neumann no perÃodo de 1929-1932, mas outras provas surgiram ao longo dos anos. A prova contida neste trabalho à de Edward Brian Davies(1994), o qual conseguiu, na prova da versÃo do teorema para cÃlculos funcionais, explicitar uma fÃrmula para f(H) (onde H à um operador nÃo-limitado e autoadjunto) para uma grande classe de funÃÃes e nÃo apenas mostrar a existÃncia do mesmo. A principal idÃia foi originalmente dada por Heler e Strojand(1989) e utiliza em sua prova teoremas conhecidos como a FÃrmula Integral de Cauchy Generalizada, Teorema da DivergÃncia, Stone Weierstrass, Teorema de Liouville, alÃm de fatos conhecidos da teoria dos operadores lineares em espacos de Hilbert. / The Spectral Theorem is one of the most famous theorems in Functional Analysis,particularly because of the large number of proofs given to it. There are versions for bounded operators, unbounded operators, self-adjoints operators, compacts, on finite-dimensional spaces, on finnite-dimensional spaces. The general version was proved by Stone and Weierstrass during the period 1929-1932, but another proofs emerged over the years. The proof in this monography was given by Edward Brian Davies(1994), which gives an explicity formula for the functional calculus f(H) (where H is an self-adjoint operator) and not only proof its existence. The main idea was originally given by Heler and Strojand(1989) and in its proofs it used well-knows theorems like Stokes' Theorem,Cauchy's Integral Formula Generalized, Stone-Weierstrass, Liouville's Theorem, besides facts of the theory of linear operators on Hilbert spaces.

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sightings 1. Unitary *-module spaces Algebraic essence of adjointability on Hilbert C*-modules . . . . . 13 a) Operators on Hilbert C*-modules - Notions. . . . . . . . . . . . . . 13 b) Essential submodules and adjointability . . . . . . . . . . . . . . . . 15 c) From Hilbert C*-modules to unitary *-module spaces . . . . . . 16 2. Operators on unitary *-module spaces Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. Graph regularity Pragmatism between weak and (strong) regularity . . . . . . . . . 27 a) Types of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 b) The case C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 c) Graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Transition. Orthogonal complementability and topology Back to Hilbert C*-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Graph regular operators on Hilbert C*-modules 4. Commutative case: Operators on C_0(X) Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Interjection. Unboundedness and graph regularity . . . . . . . . . . 55 5. Relation to adjointable operators Sources of graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6. Concrete C*-algebras Association relation and affiliation relation . . . . . . . . . . . . . . . . 61 7. Examples Graph regular operators that are not regular . . . . . . . . . . . . . 67 a) Position and momentum operators as graph regular operators on a fraction algebra related to the Weyl algebra . . 67 b) A graph regular but not regular operator on the group C*-algebra of the Heisenberg group . . . . . . . . . . . . . . . 69 c) Unbounded Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . 70 8. Bounded transform The canonical regular operator associated to a graph regular operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9. Absolute value and polar decomposition . . . . . . . . . . . . . . . 79 10. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11. Special matrices of C*-algebras Counter examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Abstract and open questions . . . . . . . . . . . . . . . . . . . . . . . . . 89 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Dank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99