Ali Zamani
We introduce the $λ$-mean transform $M_λ(T)$ of a Hilbert space operator $T$ as an extension of some operator transforms based on the Duggal transform $T^D$ by $M_λ(T) := λT + (1-λ)T^D$, and present some of its essentially properties. Among other things, we obtain estimates for the operator norm and numerical radius of the $λ$-mean transform $M_λ(T)$ in terms of the original operator $T$.
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In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators …
Extension properties for monotone and sublinear operators
该文研究单调与次线性算子的延拓问题，得到若干延拓性质，推广了正算子和线性算子的已知相关结论。
In this paper we investigate the connections between the several different extensions of the concept of absolutely summing operators.
We show that Pawlucki-Plésniak operator is bounded for some non-Markov sets.
We provide an extension operator for weighted Sobolev spaces on bounded polyhedral cones $K$ involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces.
Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f\mapsto \mbox{Im}\langle f,Af\rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.
This paper is concerned with a boundedness of trace and extension operators for Besov spaces and Triebel-Lizorkin spaces on upper half space with variable exponents. To define trace and extension operators, we introduce a quarkonial decomposition for Besov spaces and Triebel-Lizorkin spaces with variable exponents on ${\mathbb R}^n$. Furthermore, we study trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents on upper half spaces ${\mathbb R}^n_+$.
In this paper we study invertible extensions of a symmetric operator in a Hilbert space $H$. All such extensions are characterized by a parameter in the generalized Neumann's formulas. Generalized resolvents, which are generated by the invertible extensions, are extracted by a boundary condition among all generalized resolvents in the Shtraus formula.
In this paper we study extensions of commuting tuples of symmetric and isometric operators to commuting tuples of self-adjoint and unitary operators. Some conditions which ensure the existence of such extensions are presented. A multidimensional analog of the Godič-Lucenko Theorem is proved. An application to a multidimensional power-trigonometric moment problem is given.
In a real Hilbert spaces H a smooth operator F is studied, whose derivative at each point of its domain is a symmetric operator. In terms of abstract boundary conditions locally self-adjoint extensions of this operator are described. We use some concepts and facts from symplectic differential geometry.
We study extension operators on Sobolev spaces with decreasing integrability on the base of set functions associated with the operator norms. Sharp necessary conditions in the terms of the generalized density condition and the terms of weak equivalence of Euclidean and intrinsic metrics are given.
Trace and extension operators for fractional Sobolev spaces with variable exponent
本文研究变指数分数阶 Sobolev 空间的迹与延拓问题，证明在特定正则性条件下存在线性延拓算子。
We define a family of linear type (D) operators for which the inverse of their maximal monotone extensions to the bidual are not of type (D) and provide an example of an operator in this family.
We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and weights are multiplication operators in $L^2$-spaces under the assumption of existence of a family of probability measures whose Radon-Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators.
We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $T^*T$ of a densely defined linear transformation $T$ and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint e…
In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.
We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.
We prove that Burenkov's Extension Operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space.
The aim of the paper is firstly to study domains of definitions in terms of boundary conditions of minimal and maximal operators, as well as selfadjoint extensions of a minimal operator associated with the fourth-order differential operator equation. Further, we give necessary and sufficient conditions for that operators to have a purely discrete or continuous spectrum, to exist extension with resolvent from $σ_p,$ study asymptotics of spectrum in case pure discrete spectrum. Finally, give the new and more general method for evaluations of regularized traces of operators with discrete spectrum…
In this work, firstly in the Hilbert space of vector-functions L^2 (H,(-\infty,a)\bup(b,+\infty)),a<b all selfadjoint extensions of the minimal operator generated by linear singular symmetric differential expression l(\cdot)=i d/dt+A with a selfadjoint operator coefficient A in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.
It is proved that every operator from a weak$^*$-closed subspace of $\ell_1$ into a space $C(K)$ of continuous functions on a compact Hausdorff space $K$ can be extended to an operator from $\ell_1$ to $C(K)$.
In this paper we extend Korovkin's theorem to the context of sequences of weakly nonlinear and monotone operators defined on certain Banach function spaces. Several examples illustrating the theory are included.
Some Generalizations of Mercer inequality and its operator extensions
本文研究了超二次函数的Mercer不等式及其算子推广，通过用正算子替换常数得到更一般的形式，并导出了Jensen算子不等式及Hermite-Hadamard型不等式。
Essential norm of the extensions of Stevic-Sharma operator on some spaces of analytic functions
本文研究Stevic-Sharma算子扩张的本质上范数，给出了从QK(p; q)和H1到加权Bloch空间的估计。
Extension of m-isometric weighted composition operators on directed graphs
该研究针对含单回路、多分支顶点的有向图，探讨了k-拟-m-等距复合算子及加权复合算子相关内容，丰富了有向图上算子理论的研究范畴。
We prove several results establishing conditions on the Banach lattices E_1,..., E_m and F so that the Aron-Berner extensions of (separately) almost Dunford-Pettis m-linear operators from E_1 x ... x E_m to F are (separately) almost Dunford-Pettis. Illustrative examples are provided.
We prove that Stein's Extension Operator preserves Sobolev-Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n-dimensional Euclidean space.
In this paper it is proved that each densely defined $J$-skew-symmetric operator (or each $J$-isometric operator with $\overline{D(A)}=\overline{R(A)}=H$) in a Hilbert space $H$ has a $J$-skew-self-adjoint (respectively $J$-unitary) extension in a Hilbert space $\widetilde H\supseteq H$. We follow the ideas of Galindo in~[A.~Galindo, On the existence of $J$-self-adjoint extensions of $J$-symmetric operators with adjoint, Communications on pure and applied mathematics, Vol. XV, 423-425 (1962)] with necessary modifications.
Given an arbitrary $p$-Banach ideal $(0 < p \leq 1)$, we ask for geometrical properties of this ideal which are sufficient (and necessary) to allow a transfer of the principle of local reflexivity to this operator class.
It is known that two Banach space operators that are Schur coupled are also equivalent after extension, or equivalently, matricially coupled. The converse implication, that operators which are equivalent after extension or matricially coupled are also Schur coupled, was only known for Fredholm Hilbert space operators and Fredholm Banach space operators with index 0. We prove that this implication also holds for Hilbert space operators with closed range, generalizing the result for Fredholm operators, and Banach space operators that can be approximated in operator norm by invertible operators. …
The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kani…
In this work, in the Hilbert space of vector-functions L^2 (H,(-\infty,a)\cup(b,+\infty)),a<b all normal extensions of the minimal operator generated by linear singular formally normal differential expression l(\cdot)=(d/dt+A_1,d/dt+A_2) with a selfadjoint operator coefficients A_1 andA_2 in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.
An extension of the $ρ$-operator radii
本文在有界线性Hilbert空间算子的$C^{\ast}$-代数上定义了一个推广算子半径的函数，并研究了其基本性质，推广了文献中的若干结果。
Let $G$ be a compact group (not necessarily abelian) and let $Φ$ be a Young function satisfying the $Δ_2$-condition. We determine the optimal domain and the associated extended operator for both Fourier transform and the convolution operator defined on the Orlicz spaces $L^Φ(G).$
Extension of $p$-compact operators in Banach spaces
本文研究了Banach空间中p-紧算子的扩展性质，探讨了在P_λ空间下算子扩展的条件及保持范数的特性，并推导了Banach空间为L_1-前dual的必要充分条件。
We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a constructive characterization of the bounded positive extendibility of these linear mappings. From this result we can characterize the compactness of the extended operators and that when the positive extensions have closed ranges. As a main application of our general extension theorem, we present some necessary and sufficient conditions that a positive functional…
We investigate the problem of existence of a bounded extension to $C(K)$ of a bounded $c_0(I)$-valued operator $T$ defined on the subalgebra of $C(K)$ induced by a continuous increasing surjection $ϕ:K\to L$, where $K$ and $L$ are compact lines. Generalizations of some of the results of [6] about extension of $c_0$-valued operators are obtained. For instance, we prove that when a bounded extension of $T$ exists then an extension can be obtained with norm at most twice the norm of $T$. Moreover, the class of compact lines $L$ for which the $c_0$-extension property is equivalent to the $c_0(I)$-…
In our article [15] description in terms of abstract boundary conditions of all $m$-accretive extensions and their resolvents of a closed densely defined sectorial operator $S$ have been obtained. In particular, if $\{\mathcal{H},Γ\}$ is a boundary pair of $S$, then there is a bijective correspondence between all $m$-accretive extensions $\tilde{S}$ of $S$ and all pairs $\langle \mathbf{Z},X\rangle$, where $\mathbf{Z}$ is a $m$-accretive linear relation in $\mathcal{H}$ and $X:\mathrm{dom}(\mathbf{Z})\to\overline{\mathrm{ran}(S_{F})}$ is a linear operator such that: \[ \|Xe\|^2\leqslant\mathrm…
In this work we present an operator $D_μ$ constructed with the help of the cyclic group set of the $r^{\small th}$ roots of unity. This operator constitute an $r$-extension of the Dunkl operator in one variable because when $r=2$ it reduces to the classical one and admits as eigenfunctions the Bessel functions of vector index early deeply studied by Klyuchantsev. This paper is argued by specific examples and contains some interesting results which are the prelude of harmonic analysis related to this operator.
In this paper, we introduce two notions of a relative operator $(α, β)$-entropy and a Tsallis relative operator $(α, β)$-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning $α$ and $β$. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.
Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed with respect to the graph norm of $A^*$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $A^*$. After this, we will express our results using the language of Gel'fand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence o…
A generalization of the well-known results of M.G. Kre\uın about the description of selfadjoint contractive extension of a hermitian contraction is obtained. This generalization concerns the situation, where the selfadjoint operator $A$ and extensions $\widetilde A$ belong to a Kre\uın space or a Pontryagin space and their defect operators are allowed to have a fixed number of negative eigenvalues. Also a result of Yu.L. Shmul'yan on completions of nonnegative block operators is generalized for block operators with a fixed number of negative eigenvalues in a Kre\uın space. This paper is a natu…
We develop general techniques and present an approach to solve the problem of constructing a maximal Banach ideal $({\frak A},{\bf A)}$ which does not satisfy a transfer of the norm estimation in the principle of local reflexivity to its norm ${\bf A}$. This approach leads us to the investigation of product operator ideals containing ${\frak L}_2$ (the collection of all Hilbertian operators) as a factor. Using the local properties of such operator ideals -- which are typical examples of ideals with property (I) and property (S) --, trace duality and an extension of suitable finite rank operato…
We study 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponents on special Lipschitz domains. These spaces are as usual defined by restriction of the corresponding spaces on $\R^n$. In this paper we give two intrinsic characterizations of these spaces using local means and the Peetre maximal operator. Further we construct a linear and bounded extension operator following the approach done by Rychkov, which at the end also turns out to be universal.
On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices
本文研究了阿基米德向量格上单参数算子半群到其完备化的扩张，证明了在正半群类中到ru-完备化扩张的存在性与唯一性，并通过证明性质(R)向量格上的扩张定理，去除了相关结论中对ru-完备性的要求。
Completion problem for extension of m- isometric weighted composition operators on directed graphs
本文研究了有向图上加权复合算子的k-准m-等距完成问题，探讨了单侧加权位移算子的扩展方法。
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。
The aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott's theorem on contractive completion of $2$ by $2$ block operator-valued matrices. To exhibit the ap…
In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE $\Rightarrow$ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compac…
This paper concerns positive domination property of compact operators on pre-Riesz spaces. The method is embedding the pre-Riesz space to the Riesz completion. It extends the order continuous norms in pre-Riesz spaces to Riesz completions. The compactness of third power of a positive operator is obtained in a pre-Riesz space which has an order unit.
Jensen's operator inequality for convexifiable functions is obtained. This result contains classical Jensen's operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young's inequality is given.
It was recently shown in [24] that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we der…
We give an explicit and versatile parametrization of all positive selfadjoint extensions of a densely defined, closed, positive operator. In addition, we identify the Friedrichs extension by specifying the parameter to which it corresponds. This is a manuscript that was circulated as the first part of the preprint "Two papers on selfadjoint extensions of symmetric semibounded operators", INCREST Preprint Series, July 1981, Bucharest, Romania, but never published. In this LaTeX typeset version, only typos and a few inappropriate formulations have been corrected, with respect to the original man…
In the paper is considered two problems on extension of operators whose range space for the first problem (or domain space for the second one) belongs to the fixed class of finite equivalence, which is generated by a given Banach space $X$. Both problems are considered in two variants: isometric and isomorphic ones. Received a full solution of the first problem and the solution, close to final one for the second problem.
Compressions of selfadjoint and maximal dissipative extensions of non-densely defined symmetric operators
该文针对无穷维可分Hilbert空间中的非稠定对称算子，研究其自伴与最大耗散延拓在定义域闭包子空间上的压缩，重点关注定义域闭包余维数为无穷的情况，得到了这类算子特征函数的新性质。
We study the problem of extension and lifting of operators belonging to certain operator ideals, as well as that of their associated polynomials and holomorphic functions. Our results provide a characterization of $\mathcal{L}_1$ and $\mathcal{L}_{\infty}$-spaces that includes and extends those of Lindenstrauss-Rosenthal \cite{LR} using compact operators and González-Gutiérrez \cite{GG} using compact polynomials. We display several examples to show the difference between extending and lifting compact (resp. weakly compact, unconditionally convergent, separable and Rosenthal) operators to opera…
In this work, firstly in the direct sum of Hilbert spaces of vector-functions $L^{2} (H,(-\infty,a_{1})) \oplus L^{2} (H,(a_{2},b_{2}))\oplus^{2} (H,(a_{3},+\infty))$, $- \infty<a_{1}<a_{2}<b_{2}<a_{3}<+\infty$ all normal extensions of the minimal operator generated by linear singular multipoint formally normal differential expression $l=(l_{1},l_{2},l_{3}),l_{k} = \frac{d}{dt}+A_{k}$ with a selfadjoint operator coefficient $A_k k=1,2,3$ in any Hilbert space $H$, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.
In this work, firstly in the direct sum of Hilbert spaces of vector-functions L^2 (H,(-{\infty},a_1)){\Box}L^2 (H,(a_2,b_2)){\Box}L^2 (H,(a_3,+{\infty})),- {\infty}<a_1<a_2<b_2<a_3<+{\infty} all selfadjoint extensions of the minimal operator generated by linear singular multipoint symmetric differential expression l=(l_1,l_2,l_3),l_k=i d/dt+A_k with a selfadjoint operator coefficient A_k k=1,2,3 in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.
The main aim of this paper is to generalize the classical concept of positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The concept of anti-duality carries an adequate structure to define positivity in a natural way, and is still general enough to cover numerous important areas where the Hilbert space theory cannot be applied. Our running example -- illustrating the applicability of the general setting to spaces bearing poor geometrical features -- comes from noncommutative inte…
Extension Operators for Fractional Sobolev Spaces on Lipschitz Submanifolds
本文研究了分数Sobolev空间在Lipschitz子流形上的扩展算子，通过局部坐标图表构造并明确了扩展领域大小对估计的影响，应用于几何简化等数值分析领域。
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