Multiplication Operators on the Iterated Logarithmic Lipschitz Spaces of a Tree

We introduce a class of iterated logarithmic Lipschitz spaces ${\mathcal{L}^{(k)}, k \in \mathbb{N}}$ , on an infinite tree which arise naturally in the context of operator theory. We characterize boundedness and compactness of the multiplication operators on ${\mathcal{L}^{(k)}}$ and provide estimates on their operator norm and their essential norm. In addition, we determine the spectrum, characterize the multiplication operators that are bounded below, and prove that on such spaces there are no nontrivial isometric multiplication operators and no isometric zero divisors.     

Multiplication Operators on the Lipschitz Space of a Tree

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L{\mathcal{L}} we call the little Lipschitz space.     

Multiplication Operators Between the Lipschitz Space and the Space of Bounded Functions on a Tree

In this paper, we characterize the bounded and the compact multiplication operators between the space of bounded functions on the set of vertices of a rooted infinite tree T and the Banach space of complex-valued Lipschitz functions on T. We also determine the operator norm and the essential norm for the bounded multiplication operators between these spaces and show that there are no isometries among such operators.     

加权Lipschitz空间上的Littlewood-Paley算子

本文研究了加权Lipschitz空间上的Littlewood-Paley算子.,证明了一个加权Lipschitz 函数在Littlewood-Paley算子下的象或者几乎处处等于无穷或者仍是一个加权Lipschitz函数.     

Composition Operators on the Lipschitz Space of a Tree

The Lipschitz space ${\mathcal{L}}$ of an infinite tree T rooted at o is defined as the space consisting of the functions ${f : T \rightarrow \mathbb{C}}$ such that $$\beta_f = {\rm sup}\{|f(v) - f(v^-)| : v \in T\backslash\{o\}, \,v^- {\rm parent \, of \,} v\}$$ is finite. Under the norm ${\|f\|_\mathcal{L} = |f(o)|+\beta_f,\mathcal{L}}$ is a Banach space. In this article, the functions φ mapping T into itself whose induced composition operator ${C_{\varphi} : f \mapsto f \circ \varphi}$ on the Lipschitz space is bounded, compact, or an isometry, are characterized. Specifically, it is shown that the symbols of the bounded composition operators are the Lipschitz maps of T into itself viewed as a metric space under the edge-counting distance. The symbols inducing compact operators have finite range while those inducing isometries on ${\mathcal{L}}$ are precisely the onto maps fixing the root and whose images of neighboring vertices coincide or are themselves neighboring vertices. Finally, the spectrum of the operators ${C_\varphi}$ that are isometries is studied in detail.     

Compact-Friendly Multiplication Operators on Banach Function Spaces

Answering a question posed by Abramovich et al. (Indag. Math.(N.S.)10 (1999), 161-171, we prove that a positive multiplication operator on an arbitrary Banach function space X is compact-friendly if and only if the multiplier is constant on a set of positive measure.     

Multiplication Operators on Invariant Subspaces of Function Spaces

Let M_φ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator M_z, we derive some spectral properties of the multiplication operator M_φ : F → F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n ∈ {1, 2, …, + ∞}.     

Partial Multiplication of Operators in Rigged Hilbert Spaces

The problem of the multiplication of operators acting in rigged Hilbert spaces is considered. This is done, as usual, by constructing certain intermediate spaces through which the product can be factorized. In the special case where the starting space is the set of C-vectors of a self-adjoint operator A, a general procedure for constructing a special family of interspaces is given. Their definition closely reminds that of the Bessel potential spaces, to which they reduce when the starting space is the Schwartz space Some applications are considered.     

对数Bloch空间上积型算子的有界性

让H(D)表示复平面C里的单位圆盘D上的所有解析函数的全体,ψ_1,ψ_2∈H(D),而φ是D到D的解析自映射.本文刻画了对数Bloch空间上积型算子T_(ψ_1,ψ2,φ)的有界性.     

Hardy-Orlicz Spaces and Their Multiplication Operators

In this paper some formulae on the relationship between Hardy and Hardy Orlicz spaces are presented, and multiplication operators on Hardy-Orlicz spaces are discussed.     

Factorization Theorems for Multiplication Operators on Banach Function Spaces

Let X Y and Z be Banach function spaces over a measure space \({(\Omega, \Sigma, \mu)}\) . Consider the spaces of multiplication operators \({X^{Y'}}\) from X into the Köthe dual Y′ of Y, and the spaces X ^Z and \({Z^{Y'}}\) defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space \({X^Z \cdot Z^{Y'} \subseteq X^{Y'}}\) that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey–Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class \({d_{p,Z}^*}\) of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.     

Summability Properties for Multiplication Operators on Banach Function Spaces

Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized Köthe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with ${x \in X}Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized K?the duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x ? X{x \in X} and y ? Y{y \in Y} .     

Bergman空间和q-Bloch空间之间的复合算子

本文讨论了Bergman空间和q-Bloch空间(小q-Bloch空间)之间的复合算子C(ψ)的有界性和紧性特征,得到了以下结论(1)C(ψ)是q-Bloch空间(小q-Bloch空间)到Bergman空间的有界算子或紧算子之充要条件;(2)C(ψ)是Bergman空间到q-Bloch空间的有界算子或紧算子之充要条件;(3)C(ψ)是Bergman空间到小q-Bloch空间的有界算子或紧算子之充要条件,还给出了算子C0的范数估计,此处C0(f)(z)=fo(ψ)(z)-f((ψ)(0)).     

Bergman空间和q-Bloch空间之间的复合算子

本文讨论了Bergman空间和q-Bloch空间(小q-Bloch空间)之间的复合算子Cφ的有界性和紧性特征,得到了以下结论:(1)Cφ是q-Bloch空间(小q-Bloch空间)到Bergman空间的有界算子或紧算子之充要条件; (2)Cφ是Bergman空间到q-Bloch空间的有界算子或紧算子之充要条件; (3)Cφ是Bergman空间到小q-Bloch空间的有界算子或紧算子之充要条件,还给出了算子 Cφ0的范数估计,此处Cφ0(f)(z)=foφ(z)-f(φ(0))．     

Hardy空间之间的加权复合算子

本文研究了复平面中单位圆盘D上不同Hardy空间之间的加权复合算子，利用Carleson测度的概念分别给出了有界或紧的加权复合算子的充分必要条件。本文也用角数的概念给出了紧加权复合算子的一个必要条件。     

HardyOrlicz空间之间的加权复合算

该文研究了复平面中单位圆盘上不同Hardy-Orlicz空间之间的加权复合算子,利用Carleson测度不等式给出了有界或紧的加权复合算子ωC_φ:N_p→N_q的特征. 作为推论得到了加权复合算子ωC_φ:N_p→N_q有界(或紧)的充分必要条件是ωC_φ:H_p→H_q是有界(或紧)的. 此外,还给出了Hardy-Orlicz空间上可逆及Fredholm复合算子的特征.     

On Toeplitz Operators Between Fock Spaces

We study some mapping properties of Toeplitz operators T _μ associated with nonnegative Borel measures μ on the complex space ${\mathbb{C}^n}$ . In particular, we describe the bounded and compact properties of T _μ acting between Fock spaces in terms of the objects t-Berezin transforms, averaging functions, and averaging sequences of μ. We also obtain an asymptotic estimate for the norms of the operators. The results extend and complete a recent work of Z. Hu and X. Lv when both the smallest and the largest Banach–Fock spaces are taken into account.     

Composition Operators Between Analytic Campanato Spaces

This note characterizes both boundedness and compactness of a composition operator between any two analytic Campanato spaces on the unit complex disk.