A. G. Ramm
Let $A(k)u(k)=f(k) (1)$ be an operator equation, $X$ and $Y$ are Banach spaces, $k\inΔ\subset\C$ is a parameter, $A(k):X\to Y$ is a map, possibly nonlinear. Sufficient conditions are given for continuity of $u(k)$ with respect to $k$. Necessary and sufficient conditions are given for the continuity of $u(k)$ with respect to $k$ in the case of linear operators $A(k)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In the context of general Banach spaces characterizations for the maximal monotonicity of operators with non-empty domain interior as well as stronger continuity properties for such operators are provided.
On automatic continuity of operators from ordered to topological vector spaces
本文研究了从有序到拓扑向量空间的算子连续性和有界性，探讨了Frechet空间到拓扑向量空间的算子自动连续性，特别研究了Levi和Lebesgue算子。
In this paper we study the Besov continuity of pseudo-differential operators on compact homogeneous manifolds $M=G/K.$ We use the global quantization of these operators in terms of the representation theory of compact homogeneous manifolds.
We recover for discrete Schrödinger operators on the lattice Z, stronger analogues of the results by Weder, Galtabiar and Yajima and by D'Ancona and Fanelli on R.
Nonlinear composition operators in bv_p spaces: continuity and compactness
本文研究 bv_p 等序列空间上非线性复合算子的逐点、一致、局部一致、Lipschitz 与 Hölder 连续性，并给出其紧性刻画。
In this paper we investigate the modulus of continuity of differential operator of order $k$, $k\in\mathbb{N}$, on the classes of functions defined on half-line that have positive non-increasing continuous majorants of functions and their higher derivatives.
We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced.
We introduce multilinear localization operators in terms of the short-time Fourier transform, and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. For such interpretation we use the kenrel theorem for the Gelfand-Shilov space. Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when res…
We describe continuity properties of the multivalued inverse of the numerical range map $f_A:x \mapsto \left\langle Ax, x \right\rangle$ associated with a linear operator $A$ defined on a complex Hilbert space $\mathcal{H}$. We prove in particular that $f_A^{-1}$ is strongly continuous at all points of the interior of the numerical range $W(A)$. We give examples where strong and weak continuity fail on the boundary and address special cases such as normal and compact operators.
In this paper we study one-dimensional Jacobi operators on the lattice with a potential given by the skew shift. We show that the large deviation theorem takes place for Diophantine frequency and sufficiently large disorder. Combining the large deviation theorem with the avalanche principle, we prove the log-Hölder continuity of the Lyapunov exponent.
An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. Algebraic characterizations of some classes of operators that emerged in this context are invented. Various examples and counterexamples illustrating the concepts of the paper are constructed by means of weighted shifts on directed trees. Generalizations of these results that cover the case of q-quasinormal operators are established.
In this work, we establish results on the continuity of strongly singular Calderón-Zygmund operators of type $σ$ on Hardy spaces $H^p(\mathbb{R}^n)$ for $0<p\leq 1$ assuming a weaker $L^{s}-$type Hörmander condition on the kernel. Operators of this type include appropriated classes of pseudodifferential operators $OpS^{m}_{σ,b}(\mathbb{R}^n)$ and operators associated to standard $δ$-kernels of type $σ$ introduced by Álvarez and Milman. As application, we show that strongly singular Calderón-Zygmund operators are bounded from $H^{p}_{w}(\mathbb{R}^n)$ to $L^{p}_{w}(\mathbb{R}^n)$, where $w$ bel…
Strongly continuous and locally equicontinuous families of operators and their relation to bi-continuity
本文研究了在序数完备Hausdorff局部凸空间上的强连续和局部等连续算子族，探讨了Saks空间中这些概念与双连续性及等紧性之间的关系，推广了已知结果至特定算子族。
It is proved that every linear biseparating map between spaces of vector-valued differentiable functions is a weighted composition map. As a consequence, such a map is always continuous.
A necessary condition on a singular kernel for the continuity of an integral operator in Hölder spaces
该研究在含非加倍测度的度量测度空间框架下，证明了奇异积分算子极大函数有界性是其在Hölder空间连续的必要条件（当算子不降低函数正则性时），并给出紧微分流形上积分算子的应用。
In this paper we consider the cone of all positive, bounded operators acting on an infinite dimensional, complex Hilbert space, and examine bijective maps that preserve absolute continuity in both directions. It turns out that these maps are exactly those that preserve singularity in both directions. Moreover, in some weak sense, such maps are always induced by bounded, invertible, linear- or conjugate linear operators of the underlying Hilbert space. Our result gives a possible generalization of a recent theorem of Molnar which characterizes maps on the positive cone that preserve the Lebesgu…
We establish continuity and Schatten-von Neumann properties for matrix operators with matrices satisfying mixed quasi-norm estimates. These considerations also include the case when the Lebesgue and Schatten parameters are allowed to stay between $0$ and $1$. We use the results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in a broad class of modulation spaces.
We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hörmander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity and Schatten-von Neumann properties to the Born-Jordan calculus.
We prove that all the composition operators $T_f(g):= f\circ g$, which take the Adams-Frazier space $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, are continuous mappings from $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, for every integer $m\geq 2$ and every real number $1\leq p<+\infty$. The same automatic continuity property holds for Sobolev spaces $W^m_p(\mathbb{R}^n)$ for $m\geq 2$ and $1\leq p<+\infty$.
Order continuity of Arens extensions of regular multilinear operators
本文研究了正则多重线性算子Arens延拓的序连续性，通过反例说明现有结果无法推广至整个双对偶乘积，并建立了使其在全双对偶乘积上序连续的条件，同时证明了延拓至少在单变量上序连续。
Introducing Statistical Operators: Boundedness, Continuity, and Compactness
本文针对统计收敛在算子理论中的研究空白，定义并研究赋范空间间算子的统计有界性、连续性与紧性，并建立其与经典算子概念的联系。
The Pseudo-Differential Operator (p.d.o.) $h_{μ,a}$ associated with the Bessel Operator involving the symbol $a(x,y)$ whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. $h_{μ,a}$ on Hankel translation $τ$ and Hankel convolution of Gevrey functions is continuous linear map into another Gevrey spaces.
Let $s\ge 1$, $ω,ω_0\in \mathscr P_{E,s}^0$, $a\in Γ_{s}^{(ω_0)}$, and let $\mathscr B$ be a suitable invariant quasi-Banach function space, Then we prove that the pseudo-differential operator $\operatorname{Op} (a)$ is continuous from $M(ω_0ω,\mathscr B )$ to $M(ω,\mathscr B )$.
In this paper the continuity of the set valued map $p\rightarrow B_{Ω,\mathcal{X},p}(r),$ $p\in (1,+\infty),$ is proved where $B_{Ω,\mathcal{X},p}(r)$ is the closed ball of the space $L_{p}\left(Ω,Σ,μ; \mathcal{X}\right)$ centered at the origin with radius $r,$ $\left(Ω,Σ,μ\right)$ is a finite and positive measure space, $\mathcal{X}$ is separable Banach space. An application to input-output system described by Urysohn type integral operator is discussed.
Let $μ$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $μ$-continuous if $μ(\partial U)=0$. In the paper the $μ$-continuity and uniform $μ$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $μ$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.
A General Framework for Operator Continuity between Riesz Spaces
本文提出一个统一框架，研究Riesz空间间基于广义序收敛的$\mathcal{FG}$-序连续算子，证明其有序有界性、模保持性、序连续性与带刻画，并给出正算子的扩张定理。
This note relates the notion of a discrete-spectrum family of Fredholm operators, recently introduced by N. Ivanov in arXiv:2111.15081, with the classical continuity properties in the space of operators.
Let a:[0,1] -> R be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator R^a of variable order a(.) as an operator from L_p[0,1] to L_q[0,1]. Our first aim is to study its continuity properties. For example, we show that R^a is always continuous in L_p[0,1] if p>1. Surprisingly, this becomes false for p=1. In order R^a to be continuous in L_1[0,1], the function a(.) has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of R^a. We characterize functions a(.) for which R^a is a compact ope…
Using concentration-compactness arguments we prove a variant of the Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.
Continuity of Metric Projection Operator from C[0, 1] onto Pn with Applications to Mordukhovich Derivatives
本文证明了从Banach空间C[0, 1]到多项式子空间Pn的度量投影算子是单值且连续的，并利用该连续性研究了其Mordukhovich导数的Gateaux方向导数、性质及不动点性质。
Composition operators on weighted Bergman spaces of the polydisc
本文研究了多项式盘上加权伯格曼空间之间的复合算子，证明了连续性的一般结果，并通过三重盘的例子展示了仅依赖导数表征连续性的困难。
The present work is devoted to an extension of the well-known Ehrling inequalities, which quantitatively characterize compact embeddings of function spaces, to more general operators. Firstly, a modified notion of continuity for linear operators, named \emph{Ehrling continuity} and inspired by the classical Ehrling inequality, is introduced, and then, a necessary and sufficient condition for Ehrling continuity is provided via arguments based on general topology. Secondly, general completely continuous operators between normed spaces are characterized in terms of (generalized) Ehrling type ineq…
We obtain order estimates of approximation of classes $B^Ω_{p,θ}$ of periodic functions of many variables in the space $L_q$ by using operators of orthogonal projection as well as linear operators subjected to some conditions.
We consider certain anisotropic translation invariant pseudodifferential operators, belonging to a class denoted by $\mathrm{op}(\mathcal{M}^λ_ψ)$, where $λ$ and $ψ=(ψ_1,\dots,ψ_n)$ are the "order" and "weight" functions, defined on $\mathbb{R}^n$, for the corresponding space of symbols. We prove that the boundedness of a suitable function $F_p\colon\mathbb{R}^n\to[0,+\infty)$, $1<p<\infty$, associated with $λ$ and $ψ$, is necessary to let every element of $\mathrm{op}(\mathcal{M}^λ_ψ)$ be a $L^p(\mathbb{R}^n)$-multiplier. Additionally, we show that some results known in the literature can be …
This paper offers a review of the results concerning localization operators on modulation spaces, and related topics. However, our approach, based on the Grossmann-Royer transform, gives a new insight and (slightly) different proofs. We define the Grossmann-Royer transform as interpretation of the Grossmann-Royer operator in the weak sense. Although such transform is essentially the same as the cross-Wigner distribution, the proofs of several known results are simplified when it is used instead of other time-frequency representations. Due to the importance of their role in applications when de…
In this paper, we consider bivariate Cheney-Sharma operators which are not the tensor product construction. Precisely, we show that these operators preserve Lipschitz condition of a given Lipschitz continuous function f and also the properties of the modulus of continuity function when f is a modulus of continuity function.
In this paper, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the norm-to-weak$^{*}$ closedness and of property (Q) for these operators (as recently proven by Voisei). Various applications and limiting examples are given.
We prove the strong continuity of spectral multiplier operators associated with dilations of certain functions on the general Hardy space $H^1_L$ introduced by Hofmann, Lu, Mitrea, Mitrea, Yan. Our results include the heat and Poisson semigroups as well as the group of imaginary powers.
Besov and Paley-Wiener spaces, Moduli of continuity and Hardy-Steklov operators associated with the group $"ax+b"$
本研究针对直线仿射变换“ax+b”李群的三种表示，刻画了Sobolev、Besov与Paley-Wiener空间的关系，构建了L₂逼近理论并证明了Jackson型不等式。
We study continuity properties on modulation spaces for $τ$-pseudodifferential operators with symbols $a$ in Wiener amalgam spaces. We obtain boundedness results for $τ\in (0,1)$ whereas, in the end-points $τ=0$ and $τ=1$, the corresponding operators are in general unbounded. Furthermore, for $τ\in (0,1)$, we exhibit a function of $τ$ which is an upper bound for the operator norm. The continuity properties of $τ$-pseudodifferential operators, for any $τ\in [0,1]$, with symbols $a$ in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on th…
The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a complex vector space, which is a generalization of the discrete-time quantum walks with constant coin matrices, are discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of the periodic unitary transition opera…
For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation.
Continuity aspects for traces of Dirichlet forms with respect to monotone weak convergence of G-Kato measures
该研究探讨满足Hardy型不等式测度下Dirichlet型迹的解析性质，证明G-Kato测度单调弱极限下谱等收敛性并给出定量估计，应用于椭圆算子平稳解连续性及拉普拉斯算子特征值逼近。
We illustrate the composition properties for an extended family of SG Fourier integral operators. We prove continuity results for operators in this class with respect to $L^2$ and weighted modulation spaces, and discuss continuity on $\mathscr{S}$, $\mathscr{S}^\prime$ and on weighted Sobolev spaces. We study mapping properties of global wave-front sets under the action of these Fourier integral operators. We extend classical results to more general situations. For example, there are no requirements of homogeneity for the phase functions. Finally, we apply our results to the study of of the pr…
In this paper we study the $ν$-continuity of the spectrum and some of its parts. We show that the approximate point spectrum $σ_{ap}$ is upper semi-$ν$-continuous at every Fredholm operator, then we give sufficient conditions to guarantee the $ν$-continuity of $σ_{ap}$. Also we show that the restriction of the Weyl spectrum on the class of essentially $G_1$ operators is $ν$-continuous. Finally, we investigate the $ν$-continuity of the spectrum on the class of $p$-hyponormal operators.
In this paper, the author obtain the continuity of a class of linear operators on variable anisotropic Hardy-Lorentz spaces. In addition, the author also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type spaces with variable exponents. This result is still new even when the exponent function $p(\cdot)$ is $p$.
In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f \mapsto \big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M: BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexample…
For families of magnetic pseudodifferential operators defined by symbols and magnetic fields depending continuously on a real parameter $ε$, we show that the corresponding family of spectra also varies continuously with $ε$.
Continuity of differential operators for nonarchimedean Banach algebras
针对非阿基米德域$K$和交换诺特巴拿赫$K$-代数$A$，研究有限生成巴拿赫$A$-模间Grothendieck意义下$K$-线性微分算子连续性，给出$K$特征为零时算子连续的充要条件：$A$每个极大理想商是$K$的有限扩张。
We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show that a) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in [0,1]$ such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in \mathbb R_+$ such that the resolvent of its generator h…
Continuity properties of the Laguerre operator and its propagator
本文研究了与Laguerre算子一般形式相关的Cauchy问题的适定性，并将其与谐振子的全局问题联系起来。通过详细分析相关传播算子的连续性性质，建立了分数傅里叶变换和分数Hankel变换等多种积分变换之间的联系。
In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectrum of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.
We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\a<1$ and $f$ is …
Automatic time continuity of positive matrix and operator semigroups
本文研究无可测性假设的正矩阵半群，证明其在原点附近有界时自动连续，并将结果推广到序列空间上的正算子半群。
Weak Operator Continuity for Evolutionary Equations
本文研究了 Picard 意义下演化方程的解算子连续性，提出了一种针对材料律的向量值全纯函数拓扑，使得解算子在弱算子拓扑下连续，并应用于时间依赖偏微分方程的非局部均匀化及细胞迁移模型。
In this paper we establish global Lp regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on $L^p(\Rn)$, $1<p<\infty$, as well as to be bounded from Hardy space $H^1(\Rn)$ to $L^1(\Rn)$. The obtained results extend local $L^p$ regularity properties of Fourier integral operators established by Seeger, Sogge and Stein (1991) as well as global $L^2(\Rn)$ results of Asada and Fujiwara (1978) and Ruzhansky and Sugimoto (2006), to the global setting of $L^p(\Rn)$. Global boundedness in weighted Sobolev spaces $W^{σ,p}_s(…
Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces
本文研究了Banach空间上线性余切的Oseledets子空间，在可逆与不可逆情形下证明了其Hölder连续性，推广了已有结果。
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we …
In this paper we address the $W^{1,1}$-continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator $\widetilde{M}$ acting on $W^{1,1}_{\rm rad}(\mathbb{R}^d)$, the subspace of $W^{1,1}(\mat…
The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $ω:Y\to\mathbb{C}$ and $Φ:Y\to X$ are such that the weighted composition operator $W_{Φ,ω}$ is continuous, when can we guarantee that both $Φ$ and $ω$ are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over (connected) complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。