Infinite Dimensional Holomorphy Research Articles

On the representation of linear functionals on hyper-ideals of multilinear operators

A standard technique in infinite dimensional holomorphy, which produced several useful results, uses the Borel transform to represent linear functionals on certain spaces of multilinear operators between Banach spaces as multilinear operators. In this paper, we develop a technique to represent linear functionals, as linear operators, on spaces of multilinear operators that are beyond the scope of the standard technique. Concrete applications to some well-studied classes of multilinear operators, including the class of compact multilinear operators, and to one new class are provided. We can see, in particular, that sometimes our representations hold under conditions less restrictive than those of the related classical ones.

Mathematical quantum Yang–Mills theory revisited

A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang-Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang-Mills spectrum is point and countable. In addition a fine structure of the spectrum is elucidated. KEYS: Millennium Yang-Mills problem; Finite propagation speed; Sobolev inequalities; Nuclear vector spaces; Infinite-dimensional holomorphy; Friedrichs operator extensions; Variational spectral principle; Symbols and spectral theory of pseudo-differential operators.

Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications

The numerical range of holomorphic mappings arises in many aspects of non-linear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper, we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, non-linear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.

Nonlinear generalized functions and jump conditions for a standard one pressure liquid–gas model

The mixture of a liquid and a gas is classically represented by one pressure models. These models are a system of PDEs in nonconservative form and shock wave solutions do not make sense within the theory of distributions: they give rise to products of distributions that are not defined within distribution theory. But they make sense by applying a theory of nonlinear generalized functions to these equations. In contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori. Jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations. The physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible, and so they do not satisfy equally well their respective state laws on the shock waves. This modelization gives well defined explicit jump conditions. The great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results. The jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions, which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results. We expose with care the mathematical originality of the theory of nonlinear generalized functions (an original abstract analysis issued by the Leopoldo Nachbin team on infinite dimensional holomorphy) that permits to state mathematically physical facts that cannot be formulated within distribution theory, and are the key for the removal of “ambiguities” that classically appear when one tries to calculate on “multiplications of distributions” that occur in the differential equations of physics.

HOLOMORPHIC SUPERPOSITION OPERATORS BETWEEN BANACH FUNCTION SPACES

AbstractWe prove that for a large class of Banach function spaces continuity and holomorphy of superposition operators are equivalent and that bounded superposition operators are continuous. We also use techniques from infinite dimensional holomorphy to establish the boundedness of certain superposition operators. Finally, we apply our results to the study of superposition operators on weighted spaces of holomorphic functions and the$F(p, \alpha , \beta )$spaces of Zhao. Some independent properties on these spaces are also obtained.

New strips of convergence for Dirichlet series

In this article we study the interplay of the theory of classical Dirichlet series in one complex variable with recent development on monomial expansions of holomorphic functions in infinitely many variables. For a given Dirichlet series we obtain new strips of convergence in the complex plane related to Bohr’s classical strips of uniform but non absolute convergence. absolute convergence of a Dirichlet series P ∞=1 a n ns on C. He made a deep connection with infinite dimensional holomorphy. His work is very much based on the fact that if p = (pk) ∞=1 is the sequence of prime numbers, then the sequence (log pk) ∞=1 is linearly independent over the field Q. Actually he associated to each series P ∞=1 a n ns a family of formal power series P α∈N (N) 0 apα pασ z α in infinitely many variables on the in

Representing measures and infinite-dimensional holomorphy

We consider some applications of the Bishop–De Leeuw Theorem about representing measures for some algebras of analytic functions on unit balls of Banach spaces. In particular, we investigate Hardy spaces H 2 associated with corresponding algebras. Some examples are considered.

Supercyclic and Hypercyclic Non-Convolution Operators on Hilbert–Schmidt Enire Functions

A continuous linear operator T : X > X is hypercyclic/supercyclic if there is a vector / such that the orbit Orb(T, /) = {Tn/}, respectively the set of scalar multiples of the orbit elements, forms a dense set in X. A famous result states that every non-scalar convolution operator on the space H{Cn) of nvariable entire functions is hypercyclic (and thus supercyclic). This result has been extended to infinite-dimensional holomorphy and Hilbert-Schmidt entire functions. On the other hand, up to now there are few 'explicit' examples of cyclic type non-convolution operators, both in finite and infinite dimensions. In this note we establish classes of hypercyclic and supercyclic non-convolution operators on the space of HilbertSchmidt entire functions. Moreover, we establish the existence of closed infinitedimensional hypercyclic/supercyclic vector manifolds for the operators in the corresponding classes.

Perturbation theory of Schrödinger operators in infinitely many coupling parameters

In this paper we study the behaviour of Hamilton operators and their spectra which depend on infinitely many coupling parameters or, more generally, parameters taking values in some Banach space. One of the physical models which motivates this framework is a quantum particle moving in a more or less disordered medium. One may, however, also envisage other scenarios where operators are allowed to depend on interaction terms in a manner we are going to discuss below. The central idea is to vary the occurring infinitely many perturbing potentials independently. As a side aspect this then leads naturally to the analysis of a couple of interesting questions of a more or less purely mathematical flavour which belong to the field of infinite-dimensional holomorphy or holomorphy in Banach spaces. In this general setting we study in particular the stability of the self-adjointness of the operators under discussion and the analyticity of eigenvalues under the condition that the perturbing potentials belong to certain classes.

Applications of Ultraproducts to Infinite Dimensional Holomorphy.

Applications of Ultraproducts to Infinite Dimensional Holomorphy.

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.

On a problem of topologies in infinite dimensional holomorphy

The authors solve an interesting open problem concerning the equivalence of the compact-open topology τ0 and the Nachbin ported topology τω on spaces of holomorphic functions. (See, for example, the book by S. Dineen [Complex analysis in locally convex spaces, North-Holland, Amsterdam, 1981; MR0640093 (84b:46050)] for background.) Let H(U) denote the space of complex-valued holomorphic functions on an open subset U of a complex Frechet-Montel space F. Ansemil and S. Ponte [Arch. Math. (Basel) 51 (1988), no. 1, 65–70; MR0954070 (90a:46109)] showed that these two topologies agree on H(U) for balanced U if and only if, for every natural number n, P(nF) is a Montel space. Using this result, they showed that for balanced open subsets U of certain non-Schwartz, Frechet-Montel spaces, τ0=τω. Earlier, J. Mujica [J. Funct. Anal. 57 (1984), no. 1, 31–48; MR0744918 (86c:46050)] had shown that τ0=τω for Frechet-Schwartz spaces. It is not hard to see that the two topologies differ if F is not Montel. The authors' counterexample is the Frechet-Montel space F of Taskinen [Studia Math. 91 (1988), no. 1, 17–30; MR0957282 (89k:46087)]. The authors observe that the complete symmetric projective tensor product Fs⊗ˆπF contains an isomorphic copy of l1. Consequently, P(2F) cannot be Montel, and the result follows.

On a Bornological Structure in Infinite-Dimensional Holomorphy

The bornology (b) of bounded subsets with respect to continuous convergence is used on spaces of holomorphic functions. It is shown that HomcoHb(U) ≅ U for a circled convex open subset U of a complete nuclear space. Exponential laws for spaces of holomorphic functions with bornological structures are proved and the connection with Colombeau's Silva holomorphic functions is established.

A general approach to infinite-dimensional holomorphy

In this paper we present a general theory for holomorphic functions which is based on continuous convergence instead of topologies. The theory can be applied to locally convex spaces and bornological spaces.

∧(α)-nuclearity in infinite dimensional holomorphy

∧(α)-nuclearity in infinite dimensional holomorphy

Holomorphic functions on strict inductive limits

We study strict inductive limits of Frechet Montel (FM) spaces and reflexive Frechet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy.

The ∂̄ equation in DFN spaces

We obtain: “Let E be a strong dual of a complex nuclear Fréchet space (a DFN space for short) and let F be a closed C ∞ form of type (0, 1) on E. Then there exists a C ∞ function f on E as the solution of ∂ ̄ f=F.” Since every dual nuclear complete locally convex space may be considered (from the viewpoint of its bounded sets) as an inductive limit of DFN spaces this result is immediately applicable to problems of infinite dimensional holomorphy in a setting that goes far beyond that of DFN spaces. Furthermore this result and a lemma used in its proof improve previous of C. J. Henrich and P. Raboin on the ∂̄ equation in Hilbert or DFN spaces.

An example of a nuclear space in infinite dimensional holomorphy

LetU be an open subset of a complex locally convex spaceE, andH(U) the space of holomorphic functions fromU toC. If the dualE′ ofE is nuclear with respect to the topology generated by the absolutely convex compact subsets ofE, then it is shown thatH(U) endowed with the compact open topology is a nuclear space. In particular, ifE is the strong dual of a Frechet nuclear space, thenH(U) is a Frechet nuclear space.

Recent developments in infinite dimensional holomorphy

Recent developments in infinite dimensional holomorphy