Some theorems on convex functions are proved and an application of these theorems in the theory of weighted Banach spaces of holomorphic functions is investigated, too. We prove that $H_v(G)$ and $H_{v_0}(G)$ are exactly the same spaces as $H_w(G)$ and $H_{w_0}(G)$ where $w$ is the smallest log-concave majorant of $v$. This investigation is based on the theory of convex functions and some specific properties of the weighted banach spaces of holomorphic functions under considaration and we do not use the theory of associated weights.
Given a symbol $φ,$ i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator $C_φ(f)=f\circφ$ defined on the Banach spaces of holomorphic functions $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol $φ$ which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behaviour of the iterates of the symbol. As an appendix, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.
In this article, we study the multi-dimensional Bohr radii of holomorphic functions defined on the Banach sequence spaces with values in the Banach spaces. For the case of finite dimensional Banach spaces, we exhibit the exact asymptotic growth of the Bohr radius. To achieve our goal in the finite case, we use $\ell_{p'}$-summability of certain coefficients of a given polynomial in terms of its uniform norm on $\ell_p^n$. The infinite case is handled using the techniques developed in recent years from the work of Defant, Maestre and Schwarting. We crucially use several properties of the symmet…
We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($Γ, D(Γ)$) is given by $Γf = gf+Gf^\prime$ with $D(Γ) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(Γ, D(Γ))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.
Let $X$ be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_n:X\to X$ with the property that, for each $f\in X$, the functions $T_n(f)$ are polynomials converging to $f$ in the norm of the space. We completely characterize those spaces $X$ that admit a linear polynomial approximation scheme. In particular, we show that it is NOT sufficient merely that polynomials be dense in $X$.
Let $B(Ω)$ be the Banach space of holomorphic functions on a bounded connected domain $Ω$ in $\mathbb C^n$, which contains the ring of polynomials on $Ω$. In this paper, we first establish a criterion for $B(Ω)$ to be reflexive via evaluation functions on $B(Ω)$, that is, $B(Ω)$ is reflexive if and only if the evaluation functions span the dual spaces $(B(Ω))^{*} $. Moreover, under suitable assumptions on $Ω$ and $B(Ω)$, we establish a characterization of the composition operator $C_φ$ to be a Fredholm operator on $B(Ω)$ via the property of the holomorphic self-map $φ:Ω\toΩ$. Our new approach …
Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces
Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）；Indian Institute of Science（印度科学研究所）；National Institute of Science Education and Research（国家科学教育研究中心）
Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some ch…
The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic mappings on homogeneous unit balls of $\C^n$ and to obtain some sharp boundary Schwarz type lemmas for holomorphic mappings in Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sh…
Chapter 1 deals with the problem of the existence of an upper/lower envelope from a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. Vector, ordinal and topological dual interpretations of the existence conditions of such envelope and the method of its construction are given. Chapter 2 presents applications to the existence of a nontrivial (pluri)subharmonic and/or (pluri) harmonic minorant for functions in domains from finite-dimensional real or complex spa…
Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space $B(S,X)$ of Borel maps. As an application we prove two theorems about bounded holomorphic functions on $C(S,X)$, one reminiscent of the Monodromy Theorem, the other of Liouville's Theorem.
We study a "$p$-powered" version $K_n^p(F(R))$ of the well-known Bohr radius problem for the family $F(R)$ of holomorphic functions $f: R\to X$ satisfying $\|f\|<\infty$, where $\|.\|$ is a norm in the function space $F(R)$, $R\subset\mathbb{C}^n$ is a complete Reinhardt domain and $X$ is a complex Banach space. For all $p>0$, we describe in full details the asymptotic behaviour of $K_n^p(F(R))$, where $F(R)$ is (a) the Hardy space of $X$-valued holomorphic functions defined in the open unit polydisk $\mathbb{D}^n$, and (b) the space of bounded $X$-valued holomorphic or complex-valued plurihar…
We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space X. The main results are for spaces of the form X = C(K).
We study the differential equation $\frac{\partial G}{\partial\bar z}=g$ with an unbounded Banach-valued Bochner measurable function $g$ on the open unit disk $\mathbb D\subset\mathbb C$. We prove that under some conditions on the growth and essential support of $g$ such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $\mathbb D$ with values in a complex commutative unital Banach algebra.
We show that for n $\ge$ 2 the space of closed n-cycles in a strongly (n -- 2)-concave complex space has a natural structure of reduced complex space locally of finite dimension and represents the functor "analytic family of n-cycles" parametrized by Banach analytic sets.
In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $\mathbb{C}^n$ under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain $ G $ of a complex Banach space $ X $ and take values from the unit disk $ \mathbb{D} $. Furthermore, as a consequence of one of this results, we obtain a refined version of the Bohr-type inequalities for harmonic functions $ f=h+\bar{g} $ defined on a balanced domain $ G\subset X $.…
The problem of holomorphic contractibilty of Teichmüller spaces T(0, n) of the punctures spheres (n > 4) arose in the 1970s in connection with solving the algebraic equations in Banach algebras. We provide a positive solution of this problem.
