Liang Hong
Fuzzy ordered linear spaces, Riesz spaces, fuzzy Archimedean spaces and $σ$-complete fuzzy Riesz spaces were defined and studied in several works. Following the efforts along this line, we define fuzzy Riesz subspaces, fuzzy ideals, fuzzy bands and fuzzy band projections and establish their fundamental properties.
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We prove that a Hilbert space frame $\fti$ contains a Riesz basis if every subfamily $\ftj , J \subseteq I ,$ is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements.
The modulus of an order bounded functional on a Riesz space is the sum of a pair of Riesz homomorphisms if and only if the kernel of this functional is a Grothendieck subspace of the ambient Riesz space. An operator version of this fact is given.
Linear combinations of exponentials $e^{iλ_kt}$ in the case where the distance between some points $λ_k$ tends to zero are studied. D. Ullrich has proved the basis property of the divided differences of exponentials in the case when the groups consist of equal number of points all of them are close enough to integers. We have generalized this result for groups with arbitrary number of close points and obtained a full description of Riesz bases of exponential divided differences. The application to a control problem is presented.
An order bounded functional on a Riesz space is a difference of Riesz homomorphisms if and only if the kernel of this functional is a Riesz subspace of the ambient Riesz space.
Riesz completions of some spaces of regular operators
本文研究正则算子空间的 Riesz 完备化问题，并按 van Haandel 的定义将其刻画为嵌入更大值域空间的显式子空间。
Let $L$ be a Riesz space with a strong unit $e>0$.$\mathfrak{\ }$We show that a unital linear functional $H:L\rightarrow \mathbb{R}$ satisfies $% H\left( u\right) \neq 0$ for any strong unit $u\in L$ if and only if $H$ acts like a Riesz homomorphism on every $e$-clean vector subspace of $L.$ We deduce that $L$ is $e$-clean if and only if any unital linear functional $% H:L\rightarrow \mathbb{R}$ such that $H\left( u\right) \neq 0$ for any strong unit $u\in L$ is a Riesz homomorphism.
G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that g-frames share many useful properties with frames. We also give generalized version of Riesz bases and orthonormal bases. As an application, we get atomic resolutions for bounded linear operators.
An algebraic Riccati equation for linear operators is studied, which arises in systems theory. For the case that all involved operators are unbounded, the existence of infinitely many selfadjoint solutions is shown. To this end, invariant graph subspaces of the associated Hamiltonian operator matrix are constructed by means of a Riesz basis with parentheses of generalised eigenvectors and two indefinite inner products. Under additional assumptions, the existence and a representation of all bounded solutions is obtained. The theory is applied to Riccati equations of differential operators.
Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover $Y$ for a pre-Riesz space $X$, we address the question how to find vector lattice covers for subspaces of $X$, such as ideals and bands. We provide conditions such that for a directed ideal $I$ in $X$ its smallest extension ideal in $Y$ is a vector lattice cover. We show a criterion for bands in $X$ and their extension bands in $Y$ as well. Moreover, we state properties of ideals and bands in $X$ which are generated by sets, an…
We shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-Riesz decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and $T_N$ is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For $T$ is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator $S $ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is Riesz. We investigate generalized Drazin-Riesz invertible operators and also, characteri…
For alpha>0 we consider the system l_k^{(alpha-1)/2}(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf =-\frac{d^2}{dx^2}f-\frac{alpha}{x}\frac{d}{dx}f+x^2 f. We define an atomic Hardy space H^1_{at}(X), which is a subspace of L^1((0,infty), x^alpha dx). Then we prove that the space H^1_{at}(X) is also characterized by the Riesz transform Rf=\sqrtπ\frac{d}{dx}L^{-1/2}f in the sense that f\in H^1_{at}(X) if and only if f,Rf \in L^1((0,infty),x^alpha dx).
Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces
本文将经典的Kolmogorov-Riesz紧性定理扩展到渐近$L_p$空间中，引入了额外的几乎等界条件，以刻画相对紧性。
In this paper we study the tensor product of two $f$-algebras. We show that the Riesz Subspace generated by a subalgebra in an $f$-algebra is an algebra in order to prove that the Riesz tensor product of two $f$-algebras has a structure of an $f$-algebra.
Following a recent idea by Ball, we introduce the notion of strongly truncated Riesz space with a suitable spectrum. We prove that, under an extra Archimedean type condition, any strongly truncated Riesz space is isomorphic to a uniformly dense Riesz subspace of a $C_{0}\left( X\right) $-space. This turns out to be a direct generalization of the classical Kakutani Representation Theorem on Archimedean Riesz spaces with strong unit. Another representation theorem on normed Riesz spaces, due to Fremlin, will be obtained as a consequence of our main result.
Weighted projections and Riesz frames
本文通过定义子空间与对角算子代数的相容性，研究加权投影并给出Riesz框架的新刻画。
We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.
To describe a set of functions, which forms a reflexive subspace B of the classical Banach space L a special function that characterizes their average integral growth is introduced. It is shown that this function essentially depends on the geometry of B. By the way, one question of la Vallee Poussin is answered. Also a short proof of the known result about the existence of an uncomplemented subspace isomorphic to the Hilbert space in every Lebesgue - Riesz space Lp (1<p<2) is obtained.
Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if $E$ is truncated Riesz space then $E\oplus\mathbb{R}$ can be equipped with a non-standard structure of Riesz space such that $E$ becomes a Riesz subspace of $E\oplus\mathbb{R}$ and the truncation of $E$ is provided by meet with $1$. In the present paper, we assume that the truncated Riesz space $…
It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz summability which is a generalization of the Cesàro summability. Therefore, we give the completeness criteria of these spaces with $c_0(X)$-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through $M^\infty_{R}\big(\sum_k T_k\big)$ which will be assumed that is complete for every $c_0(X)$-multi…
The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ φ_n \}$ and $\{ ψ_n \}$ such that $\sum_{n=0}^\infty \ip{x}{φ_n}\ip{ψ_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ φ_n \}$ and $\{ ψ_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant …
There is a natural isomorphism from image to complement of nullspace, for a bounded linear map from a real Banach space onto a closed subspace of a real Hilbert space. This generalizes Riesz representation (self-duality of Hilbert space). The isomorphism helps solve the pressure equation of fluid dynamics.
Riesz representation theorems for positive linear operators
本文将经典Riesz表示定理从正线性泛函推广到取值于偏序向量空间的正线性算子，构造了取值于扩展正锥的Borel表示测度并给出显式公式，且不要求目标空间为向量格或赋范空间。
Let $\mathfrak A$ be a maximal subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$. If every right invariant subspace of $\mathfrak A$ in the non-commutative Hardy space $H^2$ is of Beurling type, then we say $\mathfrak A$ to be type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the non-commutative $H^1$ space. We show that the right analytic Toeplitz algebra on the non-commutative Hardy space $H^p$ associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.
Shift-invariant subspaces of Sobolev type
本文综述了Sobolev型移位不变子空间的结构，并利用傅里叶乘子将相关定理推广至Sobolev型空间。
Consider a linear functional $L$ defined on the space $\mathcal D[s]$ of Dirichlet polynomials with real coefficients and the set $\mathcal D_+[s]$ of non-negative elements in $\mathcal D[s].$ An analogue of the Riesz-Haviland theorem in this context asks: What are all $\mathcal D_+[s]$-positive linear functionals $L,$ which are moment functionals? Since the space $\mathcal D[s],$ when considered as a subspace of $C([0, \infty), \mathbb R),$ fails to be an adapted space in the sense of Choquet, the general form of Riesz-Haviland theorem is not applicable in this situation. In an attempt to ans…
Abstract Indefinite Problems in Riesz Spaces with Its Applications
研究希尔伯特空间\(X\)（连续嵌入巴拿赫格\(E\)）上泛函临界点存在性，通过构造格分解获空间分解，结合下降流不变集法与莫尔斯理论得抽象不定问题多临界点存在性，最后应用于椭圆边值问题。
In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite dimensional posi…
The main themes of this survey are as follows: (a) the canonical (Riesz--Nevanlinna) factorization in various classes of analytic functions on the disk that are smooth up to its boundary, and (b) model subspaces (i.e., invariant subspaces of the backward shift) in the Hardy spaces $H^p$ and in BMOA. It is the interrelationship and a peculiar cross-fertilization between the two topics that we wish to highlight.
Let $Γ$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $Γ$ such that the Cauchy singular integral operator $S$ is bounded on each of them, and let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$. Consider the Riesz projection $P=(I+S)/2$, the corresponding Hardy type subspaces $PX$ and $PY$, and the Toeplitz operator $T(a):PX\to PY$ defined by $T(a)f=P(af)$ for a symbol $a\in M(X,Y)$. We show that if $X\hookrightarrow Y$ and $a\in M(X,Y)\setminus\{0\}$, then $T(a)\in\mathcal{L}(PX,PY)$ has a trivial kernel in $PX$ or a dense image…
Given an arbitrary finite set of data F= {f_1,..., f_m} in L2(Rd) we prove the existence and show how to construct a "small shift invariant space" that is "closest" to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of Rd containing Zd. This is important for example in situations where we need to deal with jitter error of the data. Here small means that our solution subspace should be generated by the integer translates of a small n…
We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fréchet-Urysohn in weak* topology. We also introduce another concept of perturbation called u…
For a second countable locally compact group $G$ and a closed abelian subgroup $H$, we give a range function classification of closed subspaces in $L^2(G)$ invariant under left translation by $H$. For a family $\mathscr{A} \subset L^2(G)$, this classification ties with a set of conditions under which the translations of $\mathscr{A}$ by $H$ form a continuous frame or a Riesz sequence. When $G$ is abelian, our work relies on a fiberization map; for the more general case, we introduce an analogue of the Zak transform. Both transformations intertwine translation with modulation, and both rely on …
We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an isometry intertwining the representation with the right regular representation, that we call a Helson map. We then characterize invariant subspaces using the Helson map, and provide general characterizations of Riesz and frame sequences of orbits. These results extend to the nonabelian setting several known results for abelian groups. They also extend to co…
Pervasiveness of $\mathcal{L}^r(E,F)$ in $\mathcal{L}^r(E,F^δ)$
本文研究阿基米德Riesz空间中正则算子空间在序完备化空间中的渗透性条件，证明其蕴含Riesz完备化实现、序连续算子正则部分构成带，并给出正部公式，应用于特殊空间并解决已有问题。
Let $X$ be a ball quasi-Banach function space satisfying some mild assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In this article, the authors introduce both the Hardy space $H_X(\mathbb{R}^{n+1}_+)$ of harmonic functions and the Hardy space $\mathbb{H}_X(\mathbb{R}^{n+1}_+)$ of harmonic vectors, associated with $X$, and then establish the isomorphisms among $H_X(\mathbb{R}^n)$, $H_{X,2}(\mathbb{R}^{n+1}_+)$, and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$, where $H_{X,2}(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$ are, respectively, certain subspaces o…
Based on Bohr's equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of Besicovitch, $B(\mathbb{R},\mathbb{C})$, defined in terms of polynomial approximations. From this, we show that in an important subspace $B^2(\mathbb{R},\mathbb{C})\subset B(\mathbb{R},\mathbb{C})$, where Parseval's equality and Riesz-Fischer theorem holds, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own cl…
We introduce and study multivariate generalizations of the classical BV spaces of Jordan, F. Riesz and Wiener. The family of the introduced spaces contains or is intimately related to a considerable class of function spaces of modern analysis including BMO, BV, Morrey spaces and those of Sobolev of arbitrary smoothness, Besov and Triebel-Lizorkin spaces. We prove under mild restrictions that the BV spaces of this family are dual and present constructive characterizations of their preduals via atomic decompositions. Moreover, we show that under additional restrictions such a predual space is is…
Successive Schur-Riesz Analysis for Approximation
该研究针对逼近方法中函数块冗余导致系数不唯一、检验结果悲观的问题，提出逐次Schur-Riesz分析方法，得到与层数无关的Riesz界，还给出构造性富集过程，验证了其在对角优势估计失效场景下的有效性。
The structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group is related to a notion of Hilbert modules endowed with inner products taking values in spaces of unbounded operators. A theory of reproducing systems in such modular structures is developed, providing a general framework that includes fundamental results of shift-invariant spaces. In particular, general characterizations of Riesz and frame sequences associated to group representations are provided, extending previous results for abelian groups and for cyclic subspaces of unitary r…
We study closed subspaces of $L^2(X)$, where $(X, μ)$ is a $σ$-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group $Γ$ on $X$. We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of frames and Riesz sequences in $L^2(X)$ generated by the action of the unitary representation under consideration on a countable set of functions in $L^2(X)$. Finally, closed subspaces of $L^2(G…
Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}Δ_j$, $J\subseteq\mathbb{Z}$, of segments $Δ_j=[α_j, β_j]\subset\mathbb{R}$ such that $α_{j+1}>β_j$ and $$ \inf_{j} \left(α_{j+1}-β_j\right) = d > 0. $$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\|B\|=:b<d/2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J} U_{b}(Δ_j)$ of the mutually disjoint closed $b$-neighborhoods $U_{b}(Δ_j)$ of the segments $Δ_j$ in $\mathbb{C}$. Let $Q_j…
A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality
研究在巴拿赫空间中，由阿贝尔遍历扇形算子\(A\)及无界算子\(\partial\)构成的系统，通过反向里斯估计与谱隙条件，推出庞加莱不等式及伴随散度不等式，该方法统一且灵活，能恢复并扩展相关定理，适用于多种几何情形。
Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems of reproducing kernels in a reproducing kernel Hilbert space $\mathcal{H}$ of analytic functions. We study the completeness problem for a class of spaces with a Riesz basis of reproducing kernels and for model subspaces $K_Θ$ of the Hardy space. We find a class of spaces where systems biorthogonal to complete systems of reproducing kernels are always complete…
Nonlinear frames and sparse reconstructions in Banach spaces
本文研究Banach空间中非线性框架的扩展及其在信号重建中的应用，提出稳定重建算法并探讨稀疏信号的优化框架。
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