Complex Banach Space Research Articles

Growth, distortion and coefficient bounds for Carathéodory families in [formula omitted] and complex Banach spaces

Let X be a complex Banach space with the unit ball B. The family M is a natural generalization to complex Banach spaces of the well-known Carathéodory family of functions with positive real part on the unit disc. We consider subfamilies Mg of M depending on a univalent function g. We obtain growth theorems and coefficient bounds for holomorphic mappings in Mg, including some sharp improvements of existing results. When g is convex, we study the family Rg consisting of holomorphic mappings f:B→X which have the property that the mapping Df(z)(z) belongs to Mg. Further, we consider radius problems related to the family Rg, when X is a complex Hilbert space. In particular, if X is the Euclidean space Cn, we obtain some quasiconformal extension results for mappings in Rg. We also obtain some sufficient conditions for univalence and starlikeness in complex Banach spaces.

Hypercyclicity for the Elements of the Commutant of an Operator

Given a bounded linear operator T acting on a complex Banach space, we obtain a spectral condition implying that each operator in the commutant of T different from λI has a hypercyclic multiple, and we show several examples of operators satisfying this condition. We emphasize that for some of these examples we do not have a description of the commutant of T.

On the completion problem for algebra [formula omitted

We study the completion problem to an invertible operator-valued function for the class of bounded holomorphic functions on the unit disk D⊂S with relatively compact images in the space of bounded linear operators between complex Banach spaces. In particular, we prove that in this class of functions the operator-valued corona problem and the completion problem are not equivalent, and establish an Oka-type principle asserting that the completion problem is solvable if and only if it is solvable in the class of continuous operator-valued functions on D with relatively compact images.

On a Subclass of Close-to-Convex Mappings

In this paper, the class of close-to-convex mappings of order \(\alpha \) is introduced in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space \(X\) or the unit polydisc in \(\mathbb {C}^n\) . As an application, a sharp growth theorem for close-to-convex mappings of order \(\alpha \) is obtained.

Quadratic $\rho$-functional inequalities and equations

In this paper, we solve the quadratic $\rho$-functional inequalities where $\rho$ is a fixed complex number with $|\rho| where $\rho$ is a fixed complex number with $|\rho|< \frac{1}{2}$. Furthermore, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic $\rho$-functional equations associated with the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.

Lineability and algebrability of the set of holomorphic functions with a given domain of existence

Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.

On certain functional equation related to a class of generalized inner derivations

The main purpose of this paper is to prove the following result. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X and let A (X) ⊆ L (X) be a standard operator algebra. Suppose there exists an additive mapping T : A (X)→L (X) satisfying the relation T (An) = T (A)An−1−AT(An−2)A+An−1T (A) for all A ∈ A (X) and some fixed integer n > 2 . In this case T is of the form T (A) = AB+BA for all A ∈ A (X) and some fixed B ∈ L (X) . Mathematics subject classification (2010): 16W10, 46K15, 39B05.

Surjective maps preserving local spectral radius

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. Let x0 is a nonzero fixed vector in X. We give the concrete form of every surjective map φ from B(X) into its self, such that the local spectral radius of φ(T)φ(S)+φ(R )a tx0 equals the local spectral radius of TS + R at x0. We do not assume φ to be linear, or even additive.

On the Fekete and Szegö Problem for the Class of Starlike Mappings in Several Complex Variables

LetSbe the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known resultmaxf∈S⁡a3-λa22=1+2e-2λ/(1-λ)forλ∈0, 1. We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk inCn, which satisfies a certain condition.

On functional equations related to derivations and bicircular projections

In this paper we investigate some functional equations on standard operator algebras and semiprime rings. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X and let A (X) ⊂ L (X) be a standard operator algebra. Suppose there exists a linear mapping D : A (X) → L (X) satisfying the relation D(An) = D(A)An−1 + AD(An−2)A+An−1D(A) for all A ∈ A (X), where n > 2 is some fixed integer. In this case D is of the form D(A) = [A,B] for all A ∈ A (X) and some fixed B ∈ L (X) . Some functional equations related to bicircular projections are also investigated. Mathematics subject classification (2010): 16W10, 46K15, 39B05.

Isometries on the vector valued little Bloch space

In this paper, we describe the surjective linear isometries on a vector valued little Bloch space with range space a smooth, strictly convex and reflexive complex Banach space. We also describe the hermitian operators and the generalized bi-circular projections supported by these spaces.

On geometric extension of polynomials on Banach spaces

We consider some questions related to Aron-Berner extensions of polynomials on infinitely dimensional complex Banach spaces, using natural extensions of their zeros.

Boundaries for algebras of analytic functions on function module Banach spaces

We consider the uniform algebra of continuous and bounded functions that are analytic on the interior of the closed unit ball of a complex Banach function module X. We focus on norming subsets of , i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite-dimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an -sum of infinitely many Banach spaces and further, the torus is a boundary.

Hypercyclic and topologically mixing properties of abstract timefractional equations with discrete shifts

The most valuable theoretical results about hypercyclic and topologically mixing properties of some special subclasses of the abstract time-fractional equations of the following form: Dn t u(t) +An−1D αn−1 t u(t) + · · ·+A1D α1 t u(t) = A0D α t u(t), t > 0, u(0) = uk, k = 0, · · ·, dαne − 1. (1) where n ∈ N {1}, A0, A1, · · ·, An−1 are closed linear operators acting on a separable infinite-dimensional complex Banach space E, 0 ≤ α1 < · · · < αn, 0 ≤ α < αn, and Dt denotes the Caputo fractional derivative of order α ([1]), have been recently clarified in [12]-[13]. In this paper, we continue the analysis contained in [12]-[13] by assuming that, for every j ∈ Nn−1, the operator Aj is a certain function of unilateral backward shifts acting on weighted l(C)-spaces.

Sharp distortion theorems for a subclass of close-to-convex mappings

We introduce the class of strongly close-to-convex mappings of order α in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in ℂn. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.

Lebesgue-Type Inequalities for Quasi-greedy Bases

We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x−GNx∥ and the best N-term approximation σN(x) is controlled by max{μ(N),kN}, where μ(N) and kN are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.

Maps preserving the dimension of fixed points of products of operators

In this paper, we characterize the forms of surjective maps on which preserve the dimension of the vector space containing of all fixed points of products of operators. Here, denotes the algebra of all bounded linear operators on a complex Banach space with .

Sufficient conditions for ϵ quasi-convex mappings in a complex Banach space

In this article, some sufficient conditions for ϵ quasi-convex mappings on the unit ball B in a complex Banach space are provided. From these, we may construct many concrete ϵ quasi-convex mappings on B. Some results, presented in this article, generalize the related results of earlier authors.

On the interpolation of analytic mappings

Let (E0,E1) and (H0,H1) be two pairs of complex Banach spaces densely and continuously embedded into each other, E1 ↪ E0 and H1 ↪ H0 and also let \(\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } \). By Eθ = [E0, E1]θ and Hθ = [H0, H1]θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by Bθ(0,R) we denote an open ball of radius R in the space Eθ. Let Φ: B0(0,R) → H0 be an analytic mapping taking B1(0,R) into H1, and let the estimates $$\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball Bθ(0,r) of radius r ∈ (0,R) in the space Eθ into Hθ and $$\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $$ .

Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces

AbstractLet A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(ℝ+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equationwith initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel–Lizorkin and Lebesgue vector-valued function spaces.