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In this paper, we obtain conditions of the solvability of the Riemann boundary value problem for sectionally analytic functions in multiply connected domains in Besov spaces embedded into the class of continuous functions. We indicate a new class of Cauchy-type integrals, which are continuous on a closed domain with continuous (not Hölder) density in terms of Besov spaces, and for which the Sokhotski–Plemelj formulas are valid. [ABSTRACT FROM AUTHOR]

I introduce a notion of quaternionic regularity using techniques based on hypertwined analysis, a refined version of general hypercomplex theory. In the quaternionic and biquaternionic cases, I show that hypertwined holomorphic (regular) functions admit a decomposition in a hypertwined sum of regular functions in certain subalgebras. The hypertwined quaternionic regularity lies in between slice regularity and the modified Cauchy–Fueter theories, and proves to have a direct impact on reformulations of quaternionic and spacetime algebra quantum theories. [ABSTRACT FROM AUTHOR]

We study linear and algebraic structures in sets of Dirichlet series with maximal Bohr's strip. More precisely, we consider a set M of Dirichlet series which are uniformly continuous on the right half plane and whose strip of uniform but not absolute convergence has maximal width, i.e., 1 2 . Considering the uniform norm, we show that M contains an isometric copy of ℓ 1 (except zero) and is strongly ℵ 0 -algebrable. Also, there is a dense G δ set such that any of its elements generates a free algebra contained in M ∪ { 0 } . Furthermore, we investigate M as a subset of the Hilbert space of Dirichlet series whose coefficients are square-summable. In this case, we prove that M contains an isometric copy of ℓ 2 (except zero). [ABSTRACT FROM AUTHOR]

We introduce analytic Besov-type Bergman-Orlicz spaces on the open unit disk. By using the result on a linear isometry of the analytic Besov-type space due to C. J. Kolaski, we will give the characterization for a linear isometry of this space. Furthermore we describe the formula of a surjective multiplicative isometry of this space. [ABSTRACT FROM AUTHOR]

For Fatou’s interpolation theorem of 1906 we suggest a new elementary proof. [ABSTRACT FROM AUTHOR]

In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives. [ABSTRACT FROM AUTHOR]

We introduce interpolation sets for the Zygmund class 𝒵 {\mathcal{Z}} in the unit disc of the complex plane. This space lies between the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the class of order α = 1 {\alpha=1} , whose interpolation sets are given in a different way. We prove that the interpolation sets for 𝒵 {\mathcal{Z}} are interpolation sets for the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the latter are interpolation sets for a space slightly larger than 𝒵 {\mathcal{Z}}. [ABSTRACT FROM AUTHOR]

It is shown that the Banach algebra C ub (X , d) of bounded uniformly K -valued continuous functions on a metric space (X, d) is coherent if and only if d is a uniformly discrete metric, or equivalently, if X does not contain twin sequences. The proof is based on Neville’s result that C (X , R) for a Tychonov space is coherent if and only if X is basically disconnected. Since C ub (X , d) is self-adjoint, we also include in the survey part some general results on the spectrum M(A) of general self-adjoint Banach function algebras which we need for the special case here. [ABSTRACT FROM AUTHOR]

A Furstenberg family F is a collection of infinite subsets ofthe set of positive integers such that if A ⊂ B and A ∈ F , then B ∈ F . For aFurstenberg family F , an operator T on a topological vector space X is said tobe F -transitive provided that for each non-empty open subsets U , V of X the set { n ∈ N : T n (U) ∩ V ≠ ∅ } belongs to F . In this paper, we characterize the F -transitivityof composition operator C ϕ on the space H (Ω) of holomorphic functionson a domain Ω ⊂ C by providing a necessary and sufficient condition in terms ofthe symbol ϕ . [ABSTRACT FROM AUTHOR]

We study the density of polynomials in H 2 (E , φ) , the space of square integrable functions with respect to e - φ d m and holomorphic on the interior of E in C , where φ is a subharmonic function and dm is a measure on E. We give a result where E is the union of a Lipschitz graph and a Carathéodory domain, which we state as a weighted L 2 -version of the Mergelyan theorem. We also prove a weighted L 2 -version of the Carleman theorem. [ABSTRACT FROM AUTHOR]

Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part.

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lipα (X, d) and Dales-Davie algebras D(X, M) are BSE-algebras for certain underlying plane sets X. Moreover, we investigate BSE properties of certain subalgebras of Lipα(X, d)suchas LipA (X, α), Lipn (X, α) and Lip(X, M, α). BSE properties of Bloch type spaces ℬ α and Zygmund type spaces Zα are also investigated in different cases of α ∈ ℝ. [ABSTRACT FROM AUTHOR]

We prove that, for asymptotically bounded holomorphic functions in a sector in C , an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by Fruchard and Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén–Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of Lindelöf and Valiron. [ABSTRACT FROM AUTHOR]

For a complex Banach space X with open unit ball B X , consider the Banach algebras H ∞ (B X) of bounded scalar-valued holomorphic functions and the subalgebra A u (B X) of uniformly continuous functions on B X. Denoting either algebra by A , we study the Gleason parts of the set of scalar-valued homomorphisms M (A) on A. Following remarks on the general situation, we focus on the case X = c 0 , giving a complete characterization of the Gleason parts of M (A u (B c 0)) and, among other things, showing that every fiber in M (H ∞ (B c 0)) over a point in B ℓ ∞ contains 2 c discs lying in different Gleason parts. [ABSTRACT FROM AUTHOR]

We prove that if { φ j } j is a sequence of subharmonic functions which are increasing to some subharmonic function φ in C , then the union of all the weighted Hilbert spaces H (φ j) is dense in the weighted Hilbert space H (φ) . [ABSTRACT FROM AUTHOR]

We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite codimension. Similarly, the kernel of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite dimension. [ABSTRACT FROM AUTHOR]

We show that any element of the special linear group SL2(R) is a product of two exponentials if the ring R is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to SL2(C) is not surjective. Our result extends to the linear group GL2(R), where it is sharp as well. [ABSTRACT FROM AUTHOR]

Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in SL n (R) is a product of elementary matrices with entries in R. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where R is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in GL n (R) as a product of exponentials. [ABSTRACT FROM AUTHOR]

We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a “Hermitian” product on slice regular functions which gives us the possibility to express the ∗-product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from f and g. Afterwards we are able to determine, under different assumptions, when the sum, the ∗-product and the ∗-conjugation of two slice regular functions preserve a complex slice. We also study when the ∗-power of a slice regular function has this property or when it preserves all complex slices. To obtain these results, we prove two factorization theorems: in the first one, we are able to split a slice regular function into the product of two functions: one keeping track of the zeroes and the other which is never vanishing; in the other one, we give necessary and sufficient conditions for a slice regular function (which preserves all complex slices) to be the symmetrized of a suitable slice regular one. [ABSTRACT FROM AUTHOR]

The division algorithm for ideals of algebraic power series satisfying Hironaka’s box condition is shown to be finite when expressed suitably in terms of the defining polynomial codes of the series. In particular, the codes of the reduced standard basis of the ideal can be constructed effectively. [ABSTRACT FROM AUTHOR]

Given a complex domain [ABSTRACT FROM AUTHOR]

Consider rings of single variable real analytic or complex entire functions, denoted by K 〈 z 〉 . We study “discrete z -filters” on K and their connections with the space of maximal ideals of K 〈 z 〉 . We characterize the latter as a compact T 1 space θ K of discrete z -ultrafilters on K . We show that θ K is a bijective continuous image of β K ∖ Q ( K ) , where Q ( K ) is the set of far points of β K . θ K turns out to be the Wallman compactification of the canonically embedded image of K inside θ K . Using our characterization of θ K , we derive a Gelfand–Kolmogoroff characterization of maximal ideals of K 〈 z 〉 and show that the Krull dimension of K 〈 z 〉 is at least c . We also establish the existence of a chain of prime z -filters on K consisting of at least 2 c many elements. [ABSTRACT FROM AUTHOR]

In this paper, a complete characterization of interpolating sequences for bounded functions in Dirichlet type spaces is obtained. We also give a description of finitely generated ideals which contain a Blaschkle product whose zeros form an interpolating sequence for the ring with F-property. [ABSTRACT FROM AUTHOR]

We study the rate of growth of entire functions that are distributionally irregular for the differentiation operatorD. More specifically, given   and  , where  , we prove that there exists a distributionally irregular entire function  f for the operator  D such that itsp-integral mean function   grows not more rapidly than  . This completes related known results about the possible rates of growth of such means forD-hypercyclic entire functions. It is also obtained, the existence of dense linear submanifolds of   all whose ***nonzero vectors areD-distributionally irregular and present the same kind of growth. Furthermore, theD-distributional irregularity in weighted Banach spaces of entire functions is analysed. [ABSTRACT FROM AUTHOR]

The paper clarifies some classical results on the peak points of uniform algebras. For a compact setwith connected complement, we consider the well-known uniform algebraof complex continuous onfunctions that are analytic in the interior of. By a classical result of E. Bishop, each boundary point ofis a peak point forWe present a brief survey of some classical theorems (mainly due to Bishop) which directly imply this result on the peak points of. [ABSTRACT FROM PUBLISHER]

Letbe the family of analytic functions on. Letbe an analytic self-map ofandfor. Letbe the weighted composition operator on. It is proved that ifcontains the disk algebra, then there iswithsuch thatis an automorphism ofandon. [ABSTRACT FROM AUTHOR]

We study the maximal ideal space of, whereBis the unit ball of. Following the lead of Gleason and Schark, we analyse Gleason parts, fibers, the Sĭlov boundary and other aspects of this Banach algebra. Our work here makes good use of the inner functions construction of Aleksandrov and Hakim/Løw/Sibony, particularly as formulated by Rudin (using ideas of Ryll and Wojtaszczyk). [ABSTRACT FROM AUTHOR]

We give a simple proof of an approximation lemma used in a known paper of R. Doss. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class of functions contained in the disc algebra $${\mathcal{A}(D)}$$ . We study functions $${f \in \mathcal{A}(D)}$$ which have the property that the continuous periodic function $${u = {\rm Re}f|_{\mathbb{T}}}$$ , where $${\mathbb{T}}$$ is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function $${f \in \mathcal{A}(D)}$$ has the above property. Afterwards, we strengthen this result by proving that, generically, for every function $${f \in \mathcal{A}(D)}$$ , both continuous periodic functions $${u = {\rm Re}f|_\mathbb{T}}$$ and $${\tilde{u} = {\rm Im}f|_\mathbb{T}}$$ are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire's Category Theorem. [ABSTRACT FROM AUTHOR]

In this paper we consider several conditions for sequences of points in M( H) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona $$M(H^\infty )\backslash \mathbb{D}$$ of H is a thin sequence for H, which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H necessarily satisfy the same condition. [ABSTRACT FROM AUTHOR]

Linear isometries of N( D) onto N( D) are described, where N( D), p > 1, is the set of all holomorphic functions f on the upper half plane D = { z ∈ ℂ: Im z > 0} such that sup>0 ∫ ln (1 + |( x + iy)|) dx < +∞. Our result is an improvement of the results by D.A. Efimov. [ABSTRACT FROM AUTHOR]

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $${\mathcal{M}_V = \{f \big|_V : f \in \mathcal{M}_d\}}$$ , where d is some integer or $${\infty, \mathcal{M}_d}$$ is the multiplier algebra of the Drury-Arveson space $${H^2_d}$$ , and V is a subvariety of the unit ball. For finite dimensional d it is known that, under mild assumptions, every isomorphism between two such algebras $${\mathcal{M}_V}$$ and $${\mathcal{M}_W}$$ is induced by a biholomorphism between W and V. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where V is the proper image of a finite Riemann surface. The second deals with the case where V is a disjoint union of varieties. [ABSTRACT FROM AUTHOR]

In this paper, we extend a method of Arveson (J Funct Anal 20(3):208-233, 1975) and McCullough (J Funct Anal 135(1):93-131, 1996) to prove a tangential interpolation theorem for subalgebras of H. This tangential interpolation result implies a Töplitz corona theorem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent and Wick (Complex Anal Oper Theory 3(3):729-738, 2009) and Douglas and Sarkar (Proc CRM, 2009). [ABSTRACT FROM AUTHOR]