Algebras Of Entire Functions Research Articles

Quasi-radial operators on the Fock space and C*-algebras of entire functions

In this article, we consider quasi-radial operators on the Fock space of holomorphic functions on C n that are square integrable with respect to the Gaussian measure on C n . We provide an integral representation for quasi-radial operators. As a consequence, we study several operator theoretic properties of these operators and the Toeplitz operators with quasi-radial defining symbols. In addition to this, we construct various examples of commutative C ∗ -algebras of analytic functions on C n each associated with a particular class of quasi-radial operators.

Algebras of entire functions and representations of the twisted Heisenberg group

Algebras of entire functions and representations of the twisted Heisenberg group

A 𝐶*-algebra of entire functions

We study the maximal abelian von Neumann algebra corresponding to L ∞ ( R ) L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F 2 F^2 , but it can also be realized as a function algebra contained in F 2 F^2 . This provides an interesting example of a C ∗ C^* algebra whose elements are analytic functions.

Convergent star products on cotangent bundles of Lie groups

For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T^*G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on T^*G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter hbar . This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on {mathfrak {g}}^*. The passage to the Weyl-ordered star product, i.e. the Gutt star product on T^*G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on hbar .

Algebras of entire functions containing functions of unbounded type on a Banach space

In this paper, we consider algebras of entire analytic functions which are bounded on a prescribed family of balls in a Banach space. We investigate the structures of such algebras and describe their spectra in terms of spectra of algebras of uniformly continuous analytic functions. Some partial examples are considered. In particular, we have complete descriptions of the spectra for the case of Tsirelson space and for $c_0$.

Relative (p, q)-φ order and relative (p, q)-φ type oriented some growth investigations of composite p-adic entire functions

Let is the -algebra of entire functions defined on a complete ultrametric algebraically closed field . For a p-adic entire function and for r > 0, sup{| f (x)|:|x|= r} is denoted by | f |(r), where |⋅|(r) is a multiplicative norm on . Taking φ(r): [0, +∞) → (0, +∞) as a non-decreasing unbounded function of r, in this paper we develop some results of composite p-adic entire functions in terms of their relative (p, q) – φ order and relative (p, q) - φ lower order along with relative (p, q) - φ type and relative (p, q) - φ weak type, where p, q are two positive integers.

Some growth properties of composite p-adic entire functions on the basis of their relative order and relative lower order

Let [Formula: see text] be a complete ultrametric algebraically closed field and [Formula: see text] be the [Formula: see text]-algebra of entire functions on [Formula: see text]. For [Formula: see text], [Formula: see text], we wish to introduce the notions of relative order and relative lower order of [Formula: see text] with respect to [Formula: see text]. Hence, after proving some basic results, in this paper, we estimate some growth rates of composite p-adic entire functions on the basis of their relative orders and relative lower orders.

$(p,q)$th order oriented growth measurement of composite $p$-adic entire functions

Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and let $\mathcal{A}\left(\mathbb{K}\right)$ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire function $f\in \mathcal{A}\left( \mathbb{K}\right) $ and $r>0$, we denote by $|f|\left(r\right)$ the number $\sup \left\{ |f\left( x\right) |:|x|=r\right\}$, where $\left\vert \cdot \right\vert (r)$ is a multiplicative norm on $\mathcal{A}\left( \mathbb{K}\right)$. For any two entire functions $f\in \mathcal{A}\left(\mathbb{K}\right)$ and $g\in \mathcal{A}\left(\mathbb{K}\right)$ the ratio $\frac{|f|(r)}{|g|(r)}$ as $r\rightarrow \infty $ is called the comparative growth of $f$ with respect to $g$ in terms of their multiplicative norms. Likewise to complex analysis, in this paper we define the concept of $(p,q)$th order (respectively $(p,q)$th lower order) of growth as $\rho ^{\left( p,q\right) }\left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup } \frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$ (respectively $\lambda ^{\left( p,q\right) }\left( f\right) =\underset{ r\rightarrow +\infty }{\lim \inf }\frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$), where $p$ and $q$ are any two positive integers. We study some growth properties of composite $p$-adic entire functions on the basis of their $\left(p,q\right)$th order and $(p,q)$th lower order.

Relative order and relative type based growth properties of iterated $p$ adic entire functions

Let us suppose that $\mathbb{K}$ be a complete ultrametric algebraically closed field and $\mathcal{A}\left( \mathbb{K}\right) $ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. The main aim of this paper is to study some newly developed results related to the growth rates of iterated p-adic entire functions on the basis of their relative orders, relative type and relative weak type.

Homomorphisms and functional calculus in algebras of entire functions on Banach spaces

In the paper the homomorphisms of algebras of entire functions on Banach spaces to a commutative Banach algebra are studied. In particular, it is proposed a method of constructing of homomorphisms vanishing on homogeneous polynomials of degree less or equal than a fixed number $n$.

The Range of the Restriction Map for a Multiplicity Variety in Hörmander Algebras of Entire Functions

Characterizations of interpolating multiplicity varieties for Hörmander algebras $${A_p(\mathbb{C})}$$ and $${A^0_p(\mathbb{C})}$$ of entire functions were obtained by Berenstein and Li (J Geom Anal 5(1):1–48, 1995) and Berenstein et al. (Can J Math 47(1):28–43, 1995) for a radial subharmonic weight p with the doubling property. In this note we consider the case when the multiplicity variety is not interpolating, we compare the range of the associated restriction map for two weights $${q \leq p}$$ and investigate when the range of the restriction map on $${A_p(\mathbb{C})}$$ or $${A^0_p(\mathbb{C})}$$ contains certain subspaces associated in a natural way with the smaller weight q.

Spectra of weighted (LB)-algebras of entire functions on Banach spaces

Given a decreasing family of continuous weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions V H ( X ) and V H 0 ( X ) . Motivated by recent research by D. Carando and P. Sevilla-Peris on weighted Fréchet algebras of entire functions on Banach spaces, we determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach–Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure.

URS' For Weierstrass Products without Exponential Factors

Let $ \\cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\\lim_{r \ o \\infty} \\prod_{|a_j|\\leq r}{(1- \\fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\\shadC} $ , let $ \\cal V $ be the set of $ f\\in {\\cal W} $ such that $ {\\shadC} [\\,f]\\subset {\\cal W} $ , and let $ {\\cal H} $ be the $ {\\shadC} $ -algebra of entire functions satisfying $ { {\\lim_{r\ o \\infty } } ({\\ln M(r,f) / r})=0} $ . Then $ \\cal H $ is included in $ {\\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\\in {\\shadN} ^* $ satisfy n > m S 3. Let $ a\\in {\\shadC}^* $ satisfies $ {a^n\ ot = \\fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\ eq (1+\\xi )^{n-m} (\\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\\cal V} $ . In particular this applies to functions such as sin x and cos x .

Hypercyclic and Chaotic Convolution Operators

Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A −∞ , but it need not be hypercyclic on radial weighted algebras of entire functions.

THE THEOREM ON THE LEAST MAJORANT AND ITS APPLICATIONS. I: ENTIRE AND MEROMORPHIC FUNCTIONS

The general concept of sweeping out is used to generalize the theorem of Koosis on the least superharmonic majorant in to least majorants with respect to a convex cone of functions defined in a domain in or . This generalization is applied to the description of nontrivial ideals and analytic sets of nonuniqueness of codimension 1 in algebras of entire functions, and to the representation of meromorphic functions of given growth.

Linear topological structure of closed ideals in weighted algebras of entire functions

Linear topological structure of closed ideals in weighted algebras of entire functions

Complementation of principal ideals in weighted $({\rm FN})$-algebras of entire functions.

Complementation of principal ideals in weighted $({\rm FN})$-algebras of entire functions.

Splitting of Closed Ideals in (DFN)-Algebras of Entire Functions and the Property (DN)

Splitting of Closed Ideals in (DFN)-Algebras of Entire Functions and the Property (DN)

Splitting of closed ideals in $({\rm DFN})$-algebras of entire functions and the property $({\rm DN})$

For a plurisubharmonic weight function p on cn let Ap(Cn) denote the (DFN)-algebra of all entire functions on cn which do not grow faster than a power of exp(p). We prove that the splitting of many finitely generated closed ideals of a certain type in Ap(Cn), the splitting of a weighted 0-complex related with p, and the linear topological invariant (DN) of the strong dual of Ap(Cn) are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of p, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions p, this characterization reduces to a covexity property of the inverse of p. Using these criteria, we present a wide range of examples of weights p for which the equivalences stated above hold and also where they fail. For p a nonnegative plurisubharmonic (psh) function on cn) let Ap(Cn) denote the algebra of all entire functions f such that If(Z)l 0 depending on f. Algebras of this type arise at various places in complex analysis and functional analysis, e.g. as Fourier transforms of certain convolution algebras. The structure of their closed ideals has been studied for a long time, primarily in the work of Schwartz [25], Ehrenpreis [9], Malgrange [17], and Palamodov [23] in connection with the existence and approximation of (systems of) convolution equations. The question whether a certain parameter dependence of the right-hand side of such an equation is shared also by its solutions, is closely related with the question of the existence of a continuous linear right inverse. The existence of such a right inverse is equivalent to the splitting of the closed ideal I associated to the corresponding equation. Also, since the quotient space Ap(Cn)/I is quite often identified with the space Ap(V) of holomorphic functions on the variety V of I which satisfy the restricted growth conditions, the latter question is equivalent to the existence of a linear extension operator from Ap(V) to Ap(Cn). Answers to these questions for various algebras have been given e.g. by Grothendieck (see lVeves [28]), Cohoon [7], Djakov and Mityagin [8], and Vogt [33]. The fact that for P(z) = IZl8, s > 1 all closed ideals in Ap(C) are cornplemented, was observed by Taylor [27]. Then Meise [19] extended Taylor's resultsX using a more functional analytic approach. He showed that the structural property (DN) of the strong dual Ap(Cn)b of Ap(Cn) implies that all slowly decreasing ideals Received by the editors July 29, 1986. 1980 Mathematics Subject Classification (1985 Reon). Primary 32E25, 46E25; Secondary 46A12, 32Al .

The structure of polynomial ideals in the algebra of entire functions

The structure of polynomial ideals in the algebra of entire functions