Some Factorization Properties of Composition Operators

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

We introduce and study Hardy spaces consisting of Clifford algebra-valued functions annihilated by perturbed Dirac operators in exterior uniformly rectifiable (UR) domains, and which radiate at infinity. In this context, we establish a higher dimensional version of Cauchy’s vanishing theorem, whose proof makes use of the properties of Cauchy-like operators in exterior UR domains, a sharp version of the Divergence Theorem in exterior Ahlfors regular domains, and a good understanding of the nature of various radiation conditions and properties of the far field pattern for Clifford algebra-valued null-solutions of the Helmholtz operator.

Let X be a complex Banach space and H1 the usual Hardy space. Various properties of operators L1/H 0 1 →X and, mainly, H1→X are considered, e.g. being weakly compact, Riesz representable, Dunford-Pettis. Connections with RNP resp. aRNP and with the validity of the equation\(H^1 \widehat \otimes X \cong \mathbb{H}^1 \left( X \right)\) are also studied, the latter space being an X-valued Hardy space. Whereas results for operators L1/H 0 1 →X closely resemble well-known theorems about operators L1→ X, this is not the case for operators H1→X. E.g., for “most” classical Banach spaces X it isnot true that\(H^1 \widehat \otimes X \cong \mathbb{H}^1 \left( X \right)\) (canonically).

For disposing numerous practical application problems involving expert systems, decision-making, image processing, classifications and etc, the investigations on the constructions and basic properties of n-ary aggregation operators (nAAOs) have always been a hot research topic with important research value and significance at theoretical investigations on aggregation operators (AOs). Herein, first, we propose a method for constructing nAAOs on function spaces via a family of known ones defined on a bounded poset, where those function spaces are composed by all fuzzy sets with that bounded poset as the truth values set. This method is different from the existing construction methods of nAAOs on bounded posets and provides a unified way of constructing usual nAAOs (like t-norms, uninorms, overlap functions, etc.) on function spaces via a family of known ones. Second, we present notion of representable nAAOs on function spaces and afford their equivalent characterization. Third, we discuss some vital properties of representable nAAOs on function spaces. Fourth, it is worth noticing that the obtained results cover the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when underlying bounded poset is taken as the corresponding truth values set, respectively. As a consequence, the theoretical results obtained herein have certain promotion and basic theoretical value for the mining of new potential applications of nAAOs in real problems, especially in expert systems, decision-making, image processing and etc.

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on .

The familiar Fuglede-Putnam's Theorem is as follows (see [5], [9] and [11]): If A and B are normal operators and if X is an operator such that AX = XB , then A * X = XB * .In this paper, the hypothesis on A and B can be relaxed by using a Hilbert-Schmidt operator X : Let A be a class A operator and let B * be an invertible class A operator such that AX = XB for a Hilbert-Schmidt operator X .Then A * X = XB * .As a consequence of this result, we obtain that the range of the generalized derivation induced by this class of operators is orthogonal to its kernel.Some properties of log -hyponormal operators are also given.

Cell array faults in random-access memories (RAMs) are usually represented by Mealy automata. In such a model, multiple faults should also be representable by automata; in fact, it should be possible to compute the automaton representing a multiple fault from the automata representing the single faults that make up the multiple fault. In this paper we study properties of binary composition operations on automata that are appropriate for the representation of multiple faults in RAMs. First, we derive a set of generic conditions that every composition operation must satisfy. Second, we develop a set of physical conditions that the composition must satisfy in order to apply to stuck-at, transition and coupling faults in RAMs. Third, we represent the transition table rules used by van de Goor and Smit by a composition operation and prove that this operation satisfies both the generic and physical conditions. Fourth, we point out that in some circumstances, it is appropriate to use a different composition operation (defined by us in a previous paper) to permit a different handling of coupling faults in the presence of stuck-at or transition faults. We compare and relate the properties of the two algebras.

A map of a space into itself generates weighted shift operators in function spaces on . The spectral properties of such operators are intimately connected with the dynamics of . It was known previously that the spectrum of an operator depends only on the set of invariant ergodic measures for . Conditions for the right invertibility of the operators are obtained when is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists. Bibliography: 29 titles.

Mapping properties for the homogeneous fractional integral operator $T_{{\mit \Omega},\alpha}$ on the Hardy spaces $H^p(R^n)$ are studied. Our results give the extension of Stein-Weiss and Taibleson-Weiss's results for the boundedness of the Riesz potential operator $I_{\alpha}$ on the Hardy spaces $H^p(R^n)$.

Chapter 39 Spaces of analytic functions with integral norm

We investigate the double commutant property for a composition operator ?? , induced on the Hardy space ?2(?) by a linear fractional self-map ? of the unit disk ?. Our main result is that this property always holds, except when ? is a hyperbolic automorphism or a parabolic automorphism. Further, we show that, in both of the exceptional cases, {??}′′ is the closure of the algebra generated by ?? and ?−1?, either in the weak operator topology, if ? is a hyperbolic automorphism, or surprisingly, in the uniform operator topology, if ? is a parabolic automorphism. Finally, for each type of a linear fractional mapping, we settle the question when any of the algebras involved are equal.

In this study, our main goal is to obtain approximation properties of convolution operators for multivariables via a special method which is not included in any other methods given before, also known as $P$-statistical convergence. We present the $P$-statistical rate of this approximation and provide examples of convolution operators. It is noteworthy to express that one can not approximate $f$ by earlier results for our examples. Therefore, our results fill an important gap in the existing literature. Furthermore, we also present a $P$-statistical approximation result in the space of periodic continuous functions of period $2\pi$ for short $C^*$.

In this chapter, and the next, we shall obtain several results about Walsh-Fourier series by using properties of operators which take one space of measurable functions to another. We begin with definitions and some simple properties of the class of operators we wish to use.