Timur Oikhberg, Vladimir G. Troitsky
M. Krein proved in 1948 that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T* has an eigenvector. We present generalizations of this result as well as some applications to C*-algebras, operators on l_1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras.
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
Krein-Šmul'jan Theorem Revisited
本文将 Krein-Šmul'jan 定理推广到多算子情形，给出判断有界自伴算子的实线性组合能否成为半正定算子的充分条件。
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
We discuss Lomonosov's proof of the Pontryagin-Krein Theorem on invariant maximal non-positive subspaces, prove the refinement of one theorem from \cite{OShT} on common fixed points for a group of fractional-linear maps of operator ball and deduce its consequences. Some Burnside-type counterparts of the Pontryagin-Krein Theorem are also considered.
In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus $ϕ\mapsto ϕ(N)$ which is the proper analogue of $ϕ\mapsto \int ϕ\, dE$ in the Hilbert space situation.
In this article, we derive the Helton-Howe-Carey-Pincus trace formula as a consequence of Krein's trace formula.
In this short note we present a simple counterexample to a nonlinear version of the Krein-Rutman theorem reported in [Nonlinear Anal. 11 (2007), 3084-3090]. Correct versions of this theorem, and related results for superadditive maps are also presented.
A generalized Krein-Rutman theorem for a strongly positive bounded linear operator whose spectral radius is larger than essential spectral radius is established: the spectral radius of the operator is an algebraically simple eigenvalue with strongly positive eigenvector and other eigenvalues are less than the spectral radius.
In this note we present some generalized versions of the Krein-Rutman theorem for sectorial operators. They are formulated in a fashion that can be easily applied to elliptic operators. Another feature of these generalized versions is that they contain some information on the generalized eigenspaces associated with non-principal eigenvalues, which are helpful in the study of the dynamics of evolution equations in ordered Banach spaces.
We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.
Hodge Theorem for Krein-Feller operators on compact Riemannian manifolds
本文在紧黎曼流形上定义k-形式的Krein-Feller算子，给出其具有紧预解式的充分条件，证明了对应Hodge定理，将经典Hodge定理推广至Krein-Feller算子。
A Dynamical Approach to the Perron-Frobenius Theory and Generalized Krein-Rutman Type Theorems
本文利用线性常微分方程的基本知识提出一种自包含的动力学方法研究经典Perron-Frobenius理论，并据此建立其复版本及证明广义Krein-Rutman型定理。
The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of $Φ$-convexity. Under general conditions on the class of functions $Φ$, the Krein-Milman-Ky Fan theorem asserts then, that every compact $Φ$-convex subset of a Hausdorff space, is the $Φ$-convex hull of its $Φ$-extremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of $Φ$-extr…
We extend to the framework of locally $L^0$-convex modules some results from classical convex analysis. Namely, randomized versions of Mazur lemma and Krein-Smulian theorem under mild stability properties are provided.
Denote by $Sof(G)$ the space of sofic representations of a countable group $G$. This space is known by a result of the second author, to have a convex-like structure. We show that, in this space, minimal faces are extreme points. We then construct uncountable many extreme points for $Sof(\mathbb{F}_2)$ and show that there exists a decreasing chain of closed faces with empty intersection. Finally we construct a strangely looking sofic representation in $Sof(\mathbb{F}_2)$ that we believe it is outside of the closure of the convex hull of extreme points.
We relax assumptions for a dissipative operator in Krein space to possess a maximal non-negative invariant subspace. Our main result is a generalization of a well-known Pontrjagin-Krein-Langer-Azizov theorem. Then we investigate the semigroup properties of the restriction of the operator onto the invariant subspace.
In this note we will present an extension of the Krein-Rutman theorem for an abstract nonlinear, compact, positively 1-homogeneous, monotone non-decreasing operators on a Banach space and apply the result to many nonlinear elliptic partial differential operators.
In this work we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a r…
In the present note a functional calculus $ϕ\mapsto ϕ(A)$ for self-adjoint definitizable linear relation on Krein spaces is developed. This functional calculus is the proper analogue of $ϕ\mapsto \int ϕ\, dE$ in the Hilbert space situation. It also comprises the Spectral Theorem for self-adjoint definitizable operators on Krein spaces showing the existence of spectral projections.
We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.
The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that $B_β:=A-β$ is a sectorial operator of angle less than $\fracπ{2}$ for some $β\geq 0$. If $B_β$ is dissipative in some equivalent scalar product then the Naimark-Arocena Representation Theorem is applied to obtain a Kre\uın space unitary dilation of the semigroup.
Krein-Feller operators on Riemannian manifolds: compactness of embedding and Hodge's theorem
本文研究了Riemannian流形上的Krein-Feller算子的紧致嵌入性和Hodge定理，通过比较定理将经典结果扩展到流形上，并探讨了测度的维数条件。
We use bicombings on arcwise connected metric spaces to give definitions of convex sets and extremal points. These notions coincide with the customary ones in the classes of normed vector spaces and geodesic metric spaces which are convex in the usual sense. A rather straightforward modification of the standard proof of the Krein-Mil'man Theorem yields the result that in a large class of metric spaces every compact convex set is the closed convex hull of its extremal points. The result appears to be new even for CAT(0)-spaces.
It is shown that if $X$ is a unitary operator so that a singular subspace of~$U$ is unitarily equivalent to a singular subspace of~$UX$ (or $XU$), for each unitary operator~$U$, then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman-Krein Theorem that includes the preservation of a singular spectral subspace in this context.
The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer con…
We study unitary representations of groups in Krein spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Krein subspaces and their (reproducing) kernels and a variant of Choquet's theorem.
We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling-Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results, is the fact that the right spectrum of a quaternionic linear operator and the S-spectrum coincide. Finally, we study the Krein-Langer factorization for slice hyperholomorphic generalized Sc…
Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by M.G.Krein and recently generalized to matrix systems by L.A.Sakhnovich. We prove that the continuous analog of the adjoint polynomials converges in the upper half-plane in the case of L^2 coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szego-Kolmogorov-Krein condition is satisfied. Thus we point out that Krei…
Gel'fand triples of test and generalized functionals in Gaussian spaces are constructed and characterized.
Definition. Let J be a period-2 unitary operator (some people say J is reflection operator or reflection symmetry) and U be a linear operator. If U^*JU = J (resp. U^*JU >= J) then U is said to be J-isometry (resp. J-noncontraction). If U^*JU >= J and UJU^* >= J) then U is said to be J-binoncontraction). Theorem. If every J-isometry has nontrivial positive invariant subspace then every J-noncontraction has such a subspace. Theorem. If every J-binoncontractive J-isometry has maximal positive invariant subspace then every J-noncontraction has such a subspace. The article text is the complete text…
Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds
该研究针对黎曼流形上的Kreĭn-Feller算子，在特征函数连续假设下证明了Courant节点域定理，并在不同流形区域条件下证明了其特征函数的连续性性质。
Our goal is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.
We give a survey on generalized Krein algebras $K_{p,q}^{α,β}$ and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that $K_{2,2}^{1/2,1/2}$ is a Banach algebra. Subsequently, Widom proved the strong Szegő limit theorem for block Toeplitz determinants with symbols in $(K_{2,2}^{1/2,1/2})_{N\times N}$ and later two of the authors studied symbols in the generalized Krein algebras $(K_{p,q}^{α,β})_{N\times N}$, where $λ:=1/p+1/q=α+β$ and $λ=1$. We here extend these results to $0<λ<1$. The entire paper is based on fundamental wor…
On the Krein-Rutman theorem and beyond
本文重新审视了克雷因-鲁曼理论，提供了关于正算子中特征值问题和特征向量稳定性的一般结果与构造性估计，并通过多种微分和积分算子的例子进行了说明。
Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are discussed and clarified, i.e., assumptions to the metric structure of the underlying space and boundedness assumptions. New results involving set-valued maps and the increasingly popular set relations are obtained along the way.
We explore Birkhoff-James orthogonality of two elements in a complex Banach space by using the directional approach. Our investigation illustrates the geometric distinctions between a smooth point and a non-smooth point in a complex Banach space. As a concrete outcome of our study, we obtain a new proof of the Bhatia-Šemrl Theorem on orthogonality of linear operators.
In this paper, we investigate $Φ$-maps associated to a certain type of $α$-completely positive maps. We then prove a KSGNS (Kasparov--Stinespring--Gel'fand--Naimark--Segal) type theorem for $α$-completely positive maps on Krein $C^*$-modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant $α$-completely positive map and study the structure of minimal covariant KSGNS constructions.
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…
For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols.
Lorentz-Shimogaki-Arazy-Cwikel Theorem Revisited
本文提出一种涵盖所有 $p,q\in (0,\infty]$ 范围的新方法来研究 Lorentz-Shimogaki 与 Arazy-Cwikel 定理，并借此解决了 Levitina 及最后两位作者提出的猜想。
In this note, we study maximal monotonicity of linear relations (set-valued operators with linear graphs) on reflexive Banach spaces. We provide a new and simpler proof of a result due to Brezis-Browder which states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone. We also study Fitzpatrick functions and give an explicit formula for Fitzpatrick functions of order n for monotone symmetric linear relations.
We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally,we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodym Property.
Spectrum of weighted composition operators. Part XII. Kamowitz - Scheinberg theorem revisited
本文研究含单位元交换半单 Banach 代数上加权自同构的谱，所得结果显著推广并强化了经典 Kamowitz–Scheinberg 定理。
Keldysh's theorem revisited
本文针对全纯算子逆的主部描述问题，给出矩阵情形下Keldysh定理的简化推导，并用新主部公式推广该定理，重点研究了半单情形。
Generalized gyrovector spaces revisited
本文修正了广义陀螺向量空间中Mazur-Ulam定理的一个证明。
A Kuratowski theorem revisited
本文针对Kuratowski提出的能否去掉Baire性质函数连续性定理中可分性条件的问题，在纲基框架下进行研究，推广了Kunugi的相关结果。
Training Neural Networks Using Reproducing Kernel Space Interpolation and Model Reduction
本文提出利用再生核希尔伯特空间插值训练神经网络，推广至Krein空间，并基于多维AAK定理提出延长神经网络（PNN），在噪声环境中性能优于现有方法。
In 1977, Baillon and Haddad proved that if the gradient of a convex and continuously differentiable function is nonexpansive, then it is actually firmly nonexpansive. This result, which has become known as the Baillon-Haddad theorem, has found many applications in optimization and numerical functional analysis. In this note, we propose short alternative proofs of this result and strengthen its conclusion.
A natural analogue of the Krein--Milman theorem is shown to fail for CAT(0) spaces.
Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal strictly positive and strictly negative \hat{V}^{\pm 1}-{invariant subspaces} L_{+}, L_{-}, such that they are mutually J-orthogonal and L_{+} + L_{-} is dense in the space. (d_1) if L_1 is non-zero \hat{V}-invariant subspace, then r(\hat{V}|L_1)=r(\hat{V}) (d_2) if L_2 is non-zero \hat{V}^{-1}-invariant subspace, then r(\hat{V}^{-1}|L_2)=r(\hat{V}^{-1}) .
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.
Classically, Tannaka-Krein duality allows us to reconstruct a (co)algebra from its category of representation. In this paper we present an approach that allows us to generalise this theory to the setting of Banach spaces. This leads to several interesting applications in the directions of analytic quantum groups, bounded cohomology and galois cohomology. A large portion of this paper is dedicated to such examples.
We generalize Bourgain-Brezis-Mironescu's asymptotic formula for fractional Sobolev functions, in the setting of abstract metric measure spaces, under the assumption that at almost every point the tangent space in the measured Gromov-Hausdorff sense is a finite dimensional Banach space or a Carnot group. Our result not only covers the known results concerning Euclidean spaces, weighted Riemannian manifolds and finite dimensional Banach spaces, but also extends Bourgain-Brezis-Mironescu's formula to $\rcdkn$ spaces and sub-Riemannian manifolds.
Given a positive function $f$ on $(0,\infty)$ and a non-zero real parameter $θ$, we consider a function $I_f^θ(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^θ(X)$ in three matrices $A,B>0$ and $X$. In the literature $θ=\pm1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^θ$, thus unifying some known results for various quantum quantities.
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。