Algebra Of Analytic Functions Research Articles

Von Neumann algebras of analytic functions on the unit ball

This paper is an attempt to answer the challenge posed by Ma and Zhu in [3]. We provide examples of von Neumann algebras of holomorphic functions over the unit disc as well as over the unit ball. In the end, we introduce a novel multiplication operation on the set of all holomorphic functions over the unit disc, ensuring that this collection forms a ⁎-algebra of analytic functions over D.

Stationary phase method, powers of functions, and applications to functional analysis

The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).

Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces

Nonclassical Differentiation in the Algebras of Analytic Functions of Bounded Type

We study the vector-valued homomorphisms of the algebra of analytic functions of bounded type defined on the tensor product of a commutative Banach algebra and a Banach space. We construct the nonclassical differentiations of the indicated algebra and establish the relationship between these differentiations and the corresponding homomorphisms.

Note on bases in algebras of analytic functions on Banach spaces

Let $\{P_n\}_{n=0}^\infty$ be a sequenceof continuous algebraically independent homogeneous polynomials on a complex Banach space $X.$ We consider the following question: Under which conditions polynomials $\{P_1^{k_1}\cdots P_n^{k_n}\}$ form a Schauder (perhaps absolute) basis in the minimal subalgebra of entire functions of bounded type on $X$ which contains the sequence $\{P_n\}_{n=0}^\infty$? In the paper we study the following examples: when $P_n$ are coordinate functionals on $c_0,$ and when $P_n$ are symmetric polynomials on $\ell_1$ and on $L_\infty[0,1].$ We can see that for some cases $\{P_1^{k_1}\cdots P_n^{k_n}\}$ is a Schauder basis which is not absolute but for some cases it is absolute.

On continuity of algebraic operations in the Gelfand topologies generated by algebras of analytic functions on Banach spaces

On continuity of algebraic operations in the Gelfand topologies generated by algebras of analytic functions on Banach spaces

Spectra of weighted composition operators on algebras of analytic functions on banach spaces

Let E be a complex Banach space, with the unit ball BE. We study the spectrum of a bounded weighted composition operator uCφ on H∞(BE) determined by an analytic symbol φ with a fixed point in BE such that φ(BE) is a relatively compact subset of E, where u is an analytic function on BE.

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C ⁎ -envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H ∞ ( B E ) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.

Spectra of composition operators on algebras of analytic functions on Banach spaces

Let E be a Banach space, with unit ball BE. We study the spectrum and the essential spectrum of a composition operator on H∞(BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

A CHARACTERIZATION OF COMPOSITION OPERATORS ON ALGEBRAS OF ANALYTIC FUNCTIONS

AbstractWe give a characterization of composition operators between algebras of analytic functions on a Banach space. We show (under fairly general conditions) that they are precisely the multiplicative operators that are transposes of operators of between the preduals of the algebras. The special cases of $H^\infty(U)$ and $H_{\mathrm{b}}(U)$ are considered. In these cases, the composition operators are those which are pointwise-to-pointwise continuous and/or $\tau_0$-to-$\tau_0$ continuous (where $\tau_0$ is the compact-open topology). We obtain Banach–Stone-type theorems for these algebras.

Feynman’s Operational Calculi: Spectral Theory for Noncommuting Self-adjoint Operators

The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for any finite number of noncommuting (i.e. not necessarily commuting) bounded, self-adjoint operators A1,..., An and associated continuous Borel probability measures μ1, ⋯, μn on [0,1]. Fix A1,..., An. Then each choice of an n-tuple \(\left(\mu_1,\ldots,\mu_n\right)\) of measures determines one of Feynman’s operational calculi acting on a certain Banach algebra of analytic functions even when A1, ..., An are just bounded linear operators on a Banach space. The Hilbert space setting along with self-adjointness allows us to extend the operational calculi well beyond the analytic functions. Using results and ideas drawn largely from the proof of our main theorem, we also establish a family of Trotter product type formulas suitable for Feynman’s operational calculi.

Minimum moduli of weighted composition operators on algebras of analytic functions

We study the minimum moduli of weighted composition operators on the disk algebra and the space of bounded analytic functions.

Spectra of algebras of entire functions on Banach spaces

We obtain an explicit description of the spectrum (set of closed maximal ideals) of H b ( X ) , H_b(X), algebra of analytic functions on a Banach space X X which are bounded on bounded subsets. We show that the spectrum of H b ( X ) H_b(X) admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball B ⊂ X B\subset X are indicated.

On a cosine functional equation for operators on the algebra of analytic functions in a domain

In this paper, we give a complete characterization of all linear functionals M and L on H(G), the algebra of analytic functions on an open set G, satisfying the functional equation M(fg) = M(f)M(g) —L(f)L(g).

Th�or�me de division avec des conditions au bord

Let Ω be an open subset of ℝn and\(\mathcal{O}(\Omega )\) be a subalgebra of the algebra of analytic functions on Ω. We suppose that\(\mathcal{O}(\Omega )\) satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of\(\mathcal{O}(\Omega )\). We also suppose that\(\mathcal{O}(\Omega )\) satifies global £ojasiewicz's inequalities. We prove the following: Let\(f_{1, \ldots ,} f_p \in \mathcal{O}(\Omega )\) andf C∞ on\(\overline \Omega \) flat on ∂Ω; if for eacha ∈ Ω the Taylor's serie off ata, Taf, is in the ideal generated byTaf1,...,Tafp in the ring of formal power series, then there existα1,...,αp,C∞ on\(\overline \Omega \) flat on ∂Ω such that\(f = \sum\nolimits_{j = 1}^p {\alpha _j f_j } \). This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.

Positivity of Dunkl’s intertwining operator

For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is - under weak assumptions - intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functions. This result in particular implies that the generalized exponential kernel of the Dunkl transform is positive-definite.

Hochschild cohomology groups of certain algebras of analytic functions with coefficients in one-dimensional bimodules

We compute the algebraic and continuous Hochschild cohomology groups of certain Fréchet algebras of analytic functions on a domain U in $ℂ^n$ with coefficients in one-dimensional bimodules. Among the algebras considered, we focus on A=A(U). For this algeb

Strictly analytic functions on p-adic analytic open sets

Let $K$ be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in $K$, but when $K$ is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) \cite{2}, \cite{13} and \cite{17}. Here, we suppose the field $K$ topologically separable (example $\Bbb C_p$). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of $T$-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets $D$, and a class of admissible sets making a covering of such a $D$, so that we obtain a sheaf on $D$. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets.

Closed ideals and the Bennett-Gilbert conjecture in Banach algebras of analytic functions

Let \( {\cal B} \) be an isometric function algebra on the unit circle \( {\Bbb T} \) and let \( {\cal B}^+={\cal B}\cap {\cal A}(\overline {\Bbb D})\) be the corresponding algebra of analytic functions (where \( {\cal A}(\overline {\Bbb D}) \) is the disc algebra). For a closed ideal I in \( {\cal B}^+, \) let \( h(I)=\{z\in {\overline {\Bbb D}}:f(z)=0 \) for every \( f \in I\} \) be the hull of I and let QI be the greatest common divisor of the inner parts of the non-zero functions in I. Moreover, denote by \( I^{\cal B} \) the closed ideal in \( {\cal B} \) generated by I. We confirm the Bennett-Gilbert conjecture¶¶\( I=Q_I{{\cal A}}(\overline {\Bbb D})\cap I^{\cal B} \)¶¶under the assumption that \( h(I)\cap {\Bbb T} \) is contained in the Cantor set. This generalizes work of Esterle, Strouse and Zouakia.