Sur le calcul fonctionnel holomorphe de L. Waelbroeck

We construct a holomorphic functional calculus ofn variables in q-algebras in the sense of L. Waelbroeck [18] and extend to these algebras the holomorphic functional calculus of Arens-Calderon.

     

Sur le calcul fonctionnel holomorphe de L. Waelbroeck

We construct a holomorphic functional calculus ofn variables in q-algebras in the sense of L. Waelbroeck [18] and extend to these algebras the holomorphic functional calculus of Arens-Calderon.     

Analyticity and naturality of the multi-variable functional calculus

Mackey-complete complex commutative continuous inverse algebras generalize complex commutative Banach algebras. After constructing the Gelfand transform for these algebras, we develop the functional calculus for holomorphic functions on neighbourhoods of the joint spectra of finitely many elements and for holomorphic functions on neighbourhoods of the Gelfand spectrum. To this end, we study the algebra of holomorphic germs in weak^*${\mathrm{weak}}^{*}$-compact subsets of the dual. We emphasize the simultaneous analyticity of the functional calculus in both the function and its arguments and its naturality. Finally, we treat systems of analytic equations in these algebras.     

Calcul Fonctionnel Holomorphe Global Unicité et Invariance Par Les Homomorphismes D’algebres de Banach

LetA be a commutative Banach algebra with unit. Denote byX _A, the global spectrum ofA. There is a holomorphic functional calculusθ _A:O(X _A)→A such thatθ _A(a)=a. In this paper, we show the uniqueness of the global holomorphic functional calculus and we establish its compatibility with Banach algebra morphisms. We also extend this holomorphic functional calculus to the case ofImc algebras.      

Calcul fonctionnel holomorphe en dimension infinie dans leslmca

We construct the infinite dimensional holomorphic functional calculus in the complete locallym-convexe algebra. We also establish its stability propreties and its compatibility with continuous algebra morphisms.     

A Version of the Bartle–Graves Theorem for Temperate Functions

We prove a version of the Bartle–Graves theorem for temperate functions in the category of the b-spaces of L. Waelbroeck. As a consequence, we give a characterization of some spaces of functions with values in quotients which appear in L. Waelbroeck's holomorphic functional calculus.     

Bicomplex holomorphic functional calculus

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus.     

Calcul Fonctionnel Holomorphe Global Unicité et Invariance Par Les Homomorphismes D’algebres de Banach

LetA be a commutative Banach algebra with unit. Denote byX _A, the global spectrum ofA. There is a holomorphic functional calculusθ _A:O(X _A)→A such thatθ _A(â)=a. In this paper, we show the uniqueness of the global holomorphic functional calculus and we establish its compatibility with Banach algebra morphisms. We also extend this holomorphic functional calculus to the case ofImc algebras.     

Functional calculus for generators of uniformly bounded holomorphic semigroups

We present a method for constructing a functional calculus for (possibly unbounded) operators that generate a uniformly bounded holomorphic semigroup, e^−zA. (A will be called a generator.) These are closed, densely defined operators whose spectrum and numerical range are contained in [0,∞), with respect to an equivalent norm.     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

Some properties of analytic maps of operators inJ *-algebras

HereJ ^*-algebras are considered, i.e. linear spaces of operators mapping one complex Hilbert space into another, which have a kind of Jordan triple product structure. Balls are determined which contain the sets of values of functionalsf(S) (S any fixed operator) defined on the classes of (Fréchet-)holomorphic mapsf of the unit ball into the generalized upper half-plane and of the unit ball into the unit ball, respectively (see Theorems 1 and 2). Similar results were obtained for holomorphic maps of operators in the sense of functional calculus (see Theorems 3–5).     

A class of singular integrals on then-complex unit sphere

The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szegö, kemel and the Cauchy singular integral operator.     

On continuity of solutions for parabolic control systems and input-to-state stability

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.     

Invariance spectrale des algèbres d'opérateurs pseudodifférentiels

We construct and study several algebras of pseudodifferential operators that are closed under holomorphic functional calculus. This leads to a better understanding of the structure of inverses of elliptic pseudodifferential operators on certain non-compact manifolds. It also leads to decay properties for the solutions of these operators. To cite this article: R. Lauter et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1095–1099.     

One-parameter groups of regular quasimultipliers

Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution and integrated groups, holomorphic semigroups, and functional calculi. Some convolution Banach algebras of functions with derivatives to fractional order are needed, which we construct using the Weyl fractional calculus.     

On Some Functions That Send Each Generator of a C0-Semigroup to the Generator of a Holomorphic Semigroup

We give some conditions on functions of the Schoenberg class T for them to send the generators of uniformly bounded semigroup of class C ₀ to the generators of holomorphic semigroups. This generalizes Yosida, Balakrishnan, and Kato's result relating to fractional powers of operators. The functional calculus of generators of C ₀-semigroups which uses the class T was constructed in the preceding articles of the author.     

On Some Functions That Send Each Generator of a C0-Semigroup to the Generator of a Holomorphic Semigroup

We give some conditions on functions of the Schoenberg class T for them to send the generators of uniformly bounded semigroup of class C ₀ to the generators of holomorphic semigroups. This generalizes Yosida, Balakrishnan, and Kato's result relating to fractional powers of operators. The functional calculus of generators of C ₀-semigroups which uses the class T was constructed in the preceding articles of the author.     

Fréchet Algebra Techniques for Boundary Value Problems on Noncompact Manifolds: Fredholm Criteria and Functional Calculus via Spectral Invariance

A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven:

• 1 The algebra G of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of L(H), H a suitable Hilbert space, i. e.,
• 2 Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus.
• 3 There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and
• 4 There is a holomorphic functional calculus for the elements of G in several complex variables.

     

Joint Spectral Multipliers for Mixed Systems of Operators

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators \(L=(L_1,\ldots ,L_d);\) where some of the operators in L have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair \((\mathcal {L},A)\) are of certain weak type (1, 1). Here \(\mathcal {L}\) is the Ornstein-Uhlenbeck operator while A is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms \(A(\mathcal {L}+A)^{-1},\) \(\mathcal {L}(\mathcal {L}+A)^{-1}\).     

Infinite dimensional holomorphy via categorical differential calculus

We establish that the category of hological spaces is equipped for calculus with complex scalars. This provides a theory of infinite dimensional holomorphy which allows maps to have nonconvex domains with empty interior. Some relatively elementary functions, hitherto excluded by the restrictive definitions of other theories, emerge as holomorphic maps.KOSEF aided.NSERC aided.