An extension of Mercer's theorem to unbounded operators

Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian

We compare the space of square-summable functions on an infinite graph (denoted $\ell^2(G)$) with the space of functions of finite energy (denoted $\mathcal{H}_{\mathcal{E}}$). There is a notion of inclusion that allows $\ell^2(G)$ to be embedded into $\mathcal{H}_{\mathcal{E}}$, but the required inclusion operator is unbounded in most interesting cases. These observations assist in the construction of the Krein extension of the Laplace operator on $\mathcal{H}_{\mathcal{E}}$. We investigate the Krein extension and compare it to the Friedrichs extension developed by the authors in a previous p…

Unbounded local completely contractive maps

We prove a local convex version of Arveson's extension theorem and of Wittstock's extension theorem. Also we prove a Stinespring type theorem for unbounded local completely contractive maps.

On the Fundamental Principles of Unbounded Functional Calculi

In this paper, a new axiomatization for unbounded functional calculi is proposed and the associated theory is elaborated comprising, among others, uniqueness and compatibility results and extension theorems of algebraic and topological nature. In contrast to earlier approaches, no commutativity assumptions need to be made about the underlying algebras. In a second part, the abstract theory is illustrated in familiar situations (sectorial operators, semigroup generators, normal operators). New topological extension theorems are proved for the sectorial calculus and the Hille--Phillips calculus.…

On a Space of Infinitely Differentiable Functions on an Unbounded Convex Set in ${\mathbb R}^n$ Admitting Holomorphic Extension in ${\mathbb C}^n$ and its Dual

We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.

Douglas与Sebestyén关于无界算子定理的推广与精细化

An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators

本文通过将Sebestyén定理推广到无界算子情形，给出闭稠定算子借助有界非负算子分解的充要条件，并据此推广和细化了Douglas关于算子不等式与分解等价性的经典定理。

Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension

This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in $L^p$ for any $p \in (2, \infty ]$, there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in $L^p$ for some $p \in (2, \infty ]$. In the case when the matrix-valued truncated Toeplitz operator has a symbol in $L^p$ for some $p \in (2, \infty ]$, an approach is developed which bypasses some of the technical…

Unbounded Wiener-Hopf Operators and Isomorphic Singular Integral Operators

Some preliminaries and basic facts regarding unbounded Wiener-Hopf operators (WH) are provided. WH with rational symbols are studied in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains and ranges are explicitly determined. A further topic concerns semibounded WH. Expressing a semibounded WH by a product of a closable operator and its adjoint this representation allows for a natural self-adjoint extension. It is shown that it coincides with the Friedrichs extension. Polar decomposition gives r…

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric measure spaces

In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condit…

Adjoint Pairs and Unbounded Normal Operators

An adjoint pair is a pair of densely defined linear operators $A, B$ on a Hilbert space such that $\langle Ax,y\rangle=\langle x,By\rangle$ for $x\in \cD(A), y \in \cD(B).$ We consider adjoint pairs for which $0$ is a regular point for both operators and associate a boundary triplet to such an adjoint pair. Proper extensions of the operator $B$ are in one-to-one correspondence $T_\cC\leftrightarrow \cC $ to closed subspaces $\cC$ of $\cN(A^*)\oplus\cN(B^*)$. In the case when $B$ is formally normal and $\cD(A)=\cD(B)$, the normal operators $T_\cC$ are characterized. Next we assume that $B$ has …

On Burenkov's extension operator preserving Sobolev-Morrey spaces on Lipschitz domains

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.

On Stein's extension operator preserving Sobolev-Morrey spaces

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.

Hilbert空间上的无界反线性算子

Unbounded Antilinear Operators on Hilbert Spaces

本文引入Hilbert空间上的无界反线性算子，建立其基本理论，包括闭值域定理、极分解定理和数值范围的凸性，并给出反线性正规算子的新结果及反线性次正规算子存在最小反线性正规扩张的充要条件，最后通过Schur和二次补框架刻画了纯反线性分块算子矩阵的可闭性。

通过Banach空间中无界谱算子的函数演算构造强可导映射

Construction of strong derivable maps via functional calculus of unbounded spectral operators in Banach spaces

本文利用无界标量型谱算子的函数演算及经典结果向完备局部凸空间的推广，给出强可导映射存在的充分条件并计算其导数，进而得到收敛于完备局部凸空间扩张积分的积分序列。

Concentration in unbounded metric spaces and algorithmic stability

We prove an extension of McDiarmid's inequality for metric spaces with unbounded diameter. To this end, we introduce the notion of the {\em subgaussian diameter}, which is a distribution-dependent refinement of the metric diameter. Our technique provides an alternative approach to that of Kutin and Niyogi's method of weakly difference-bounded functions, and yields nontrivial, dimension-free results in some interesting cases where the former does not. As an application, we give apparently the first generalization bound in the algorithmic stability setting that holds for unbounded loss functions…

关于希尔伯特空间上的$C$-对称与$C$-自伴无界算子

On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space

本文研究希尔伯特空间上关于共轭$C$的$C$-对称与$C$-自伴无界算子，给出了$C$-自伴扩张的完全描述、基于拟解析向量的判据及极分解刻画。

Unbounded Toeplitz Operators with Rational Symbols

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it fu…

无界锥上MIT袋模型的自伴性

Self-adjointness for the MIT bag model on an unbounded cone

该研究针对无界三维圆锥上带MIT袋边界条件的无质量Dirac算子，证明凸锥下其在四分量$H^1$函数域上自伴，给出非凸锥情形的数值证据，还得到凸锥上的Hardy型不等式及扰动稳定性，并讨论了向量子点边界条件的推广。

Lower bounds on the norms of extension operators for Lipschitz domains

Let $Ω\subset\dR^d$ be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space $H^1(Ω)$ up to functions in the Sobolev space $H^1(\dR^d)$ via a linear operator. The minimal possible norm of such an operator is estimated from below in terms of spectral properties of self-adjoint Robin Laplacians on domains $Ω$ and $\dR^d\setminus\ovΩ$. Another estimate of this norm is also given, where spectral properties of Schrödinger operators with the $δ$-interaction supported on the hypersurface $\partialΩ$ are involved. General re…

Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics

In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators $A$ and $B$, with $A \subseteq B^\star$ and $B \subseteq A^\star$. In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We als…

Decomposition of Normal Operators and Its Application to Spectral Theorem

A decomposition theorem for self-adjoint operators proved by Riesz and Lorch is extended to normal operators. This extension gives a new proof of the spectral theorem for unbounded normal operators.

Lattice copies and applications for weak L- and M-weakly compact operators on Banach lattices

Several recent papers investigated lattice copies and unbounded convergences in Banach lattices. In this paper, we first solve the problem of RV and LA which is an extension of the well-known James distortion theorem. Using lattice copies and unbounded convergence, then we introduce weak L- and M-weakly compact operators on Banach lattices and research the relationship between these operators and L- and M-weakly compact operators. Finally, we study the compactness of weak L-weakly compact and weak M-weakly compact operators.

Extensions of symmetric operators that are invariant under scaling and applications to indicial operators

Indicial operators are model operators associated to an elliptic differential operator near a corner singularity on a stratified manifold. These model operators are defined on generalized tangent cone configurations and exhibit a natural scaling invariance property with respect to dilations of the radial variable. In this paper we discuss extensions of symmetric indicial operators from a functional analytic point of view. In the first, purely abstract part of this paper, we consider a general unbounded symmetric operator that exhibits invariance with respect to an abstract scaling action on a …

Lower bounds for unbounded operators and semigroups

Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some estimates for the resolvent of $B$, and also to extend bounded operators related to $A$, for example a semigroup generated by $A$. When $X$ is a Hilbert space, one may also want $Y$ to be Hilbert space. Results of this type for bounded operators have been given by Arens, Read, Müller and Badea, and we give some extensions of their results to unbounded operator…

Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems

For a strongly elliptic second-order operator $A$ on a bounded domain $Ω\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(Ω)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of Hölder $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show…

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator $S$ on $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$. We compare the domain of $T$ with the domain of its adjoint $T^*$ and show that the skew part of $T$ admits two distinct linear maximal monot…

Simultaneous extension of two bounded operators between Hilbert spaces

The paper is concerned with the following question: if $A$ and $B$ are two bounded operators between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, and $\mathcal{M}$ and $\mathcal{N}$ are two closed subspaces in $\mathcal{H}$, when will there exist a bounded operator $C:\mathcal{H}\to\mathcal{K}$ which coincides with $A$ on $\mathcal{M}$ and with $B$ on $\mathcal{N}$ simultaneously? Besides answering this and some related questions, we also wish to emphasize the role played by the class of so-called semiclosed operators and the unbounded Moore-Penrose inverse in this work. Finally, we will re…

无界符号、热流与Toeplitz算子

Bounded Toeplitz operators and unbounded projected multiplication

该研究通过构造特定符号证明了Toeplitz算子在无界符号情况下与U_g算子的解耦，并揭示了热流正则性的不可逆性及无界性障碍的根源。