Dirk Werner
We prove the norm identity $\|\Id+T\| =1+\|T\|$, which is known as the Daugavet equation, for weakly compact operators $T$ on natural function spaces such as function algebras and $L^{1}$-predual spaces, provided a non-discreteness assumption is met. We also consider $c_{0}$-factorable operators and operators on $C_Λ$-spaces.
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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 find the largest linear space of bounded linear operators on L_1(Omega), that being restricted to any L_1(A), A \subset Omega, satisfy the Daugavet equation.
We introduce a substitute for the concept of slice for the case of non-linear Lipschitz functionals and transfer to the non-linear case some results about the Daugavet and the alternative Daugavet equations previously known only for linear operators.
On the weak operator Daugavet property and poor subspaces
该研究构造了具Daugavet性质但无弱算子Daugavet性质的巴拿赫空间实例，证明具Daugavet性质的可分巴拿赫空间含与ℓ¹同构的非贫子空间，且L^∞[0,1]的可分子空间均为贫的。
We show that if $T$ is a narrow operator on $X=X_{1}\oplus_{1} X_{2}$ or $X=X_{1}\oplus_{\infty} X_{2}$, then the restrictions to $X_{1}$ and $X_{2}$ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.
We prove the norm identity $\|Id+T\| = 1+\|T\|$, which is known as the Daugavet equation, for operators $T$ on $C(S)$ not fixing a copy of $C(S)$, where $S$ is a compact metric space without isolated points.
Characterizing the Daugavet property by squares of rank-one operators
本文证明维数大于1的实巴拿赫空间中秩一算子平方的范数恒等式可刻画Daugavet性质，并回答了两个开放问题，同时比较了相关逐点概念。
We study the Daugavet property in the space of Lipschitz functions $\operatorname{Lip}_0(M)$ for a complete metric space $M$. Namely we show that $\operatorname{Lip}_0(M)$ has the Daugavet property if and only if $M$ is a length space. This condition also characterises the Daugavet property in the Lipschitz free space $\mathcal{F}(M)$. Moreover, when $M$ is compact, we show that either $\mathcal{F}(M)$ has the Daugavet property or its unit ball has a strongly exposed point. If $M$ is an infinite compact subset of a strictly convex Banach space then the Daugavet property of $\operatorname{Lip}_…
We investigate the norm identity $\|uC_ϕ+ T\| = \|u\|_\infty + \|T\|$ for classes of operators on $C(S)$, where $S$ is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator $T : C(S)\to C(S)$. We also give a characterization of such weighted composition operator acting on the disk algebra $A(D).$
An operator $G : \allowbreak X \to Y$ is said to be a Daugavet center if $\|G + T\| = \|G\| + \|T\|$ for every rank-1 operator $T : \allowbreak X \to Y$. The main result of the paper is: if $G : \allowbreak X \to Y$ is a Daugavet center, $Y$ is a subspace of a Banach space $E$, and $J: Y \to E$ is the natural embedding operator, then $E$ can be equivalently renormed in such a way, that $J \circ G : X \to E$ is also a Daugavet center. This result was previously known for particular case $X=Y$, $G=\mathrm{Id}$ and only in separable spaces. The proof of our generalization is based on an idea comp…
Let $X$ be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on $X$ which depend only on the norms of images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of $X$ previously studied in the context of the classical spaces $C(K)$ and $L_1(μ)$.
Let $T\dopu C(S)\to C(S)$ be a bounded linear operator. We present a necessary and sufficient condition for the so-called Daugavet equation $$ \|\Id+T\| = 1+\|T\| $$ to hold, and we apply it to weakly compact operators and to operators factoring through $c_{0}$. Thus we obtain very simple proofs of results by Foias, Singer, Pelczynski, Holub and others.
We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor produc…
Let X be a closed subspace of a Banach space Y and J be the inclusion map. We say that the pair (X,Y) has the Daugavet property if for every rank one bounded linear operator T from X to Y the following equality \|J+T\|=1+\|T\| holds. A new characterization of the Daugavet property in terms of weak open sets is given. It is shown that the operators not fixing copies of l_1 on a Daugavet pair satisfy the Daugavet equation. Some hereditary properties are found: if X is a Daugavet space and Y is its subspace, then Y is also a Daugavet space provided X/Y has the Radon-Nikodym property; if Y is refl…
We study the presence of $L$-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that, if $\dens(Y)\leq ω_1$ and $G:X\longrightarrow Y$ is a Daugavet center, then $G(W)$ contains some $L$-orthogonal for every non-empty $w^*$-open subset of $B_{X^{**}}$. In the context of narrow operators, we show that if $X$ is separable and $T:X\longrightarrow Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^*$-open subset $W$ of $B_{X^{**}}$ then $W$ contains some $L$-orthogonal $u$ so that $T^{**}(u)=T(y)$. In the particular case that $T^*(Y^*)$ is separ…
Requirements under which the Daugavet equation and the alternative Daugavet equation hold for pairs of nonlinear maps between Banach spaces are analysed. A geometric description is given in terms of nonlinear slices. Some local versions of these properties are also introduced and studied, as well as tests for checking if the required conditions are satisfied in relevant cases.
A linear continuous nonzero operator G:X->Y is a Daugavet center if every rank-1 operator T:X->Y satisfies ||G+T||=||G||+||T||. We study the case when either X or Y is a sum $X_1 \oplus_F X_2$ of two Banach spaces $X_1$ and $X_2$ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces $X_1$ and $X_2$ there exists a Daugavet center acting from $X_1\oplus_F X_2$, and the class of those F such that for some pair of spaces $X_1$ and $X_2$ there is a Daugavet center acting into $X_1\oplus_F X_2$. We also present several examples of such Daugavet…
Weak operator Daugavet property and weakly open sets in tensor product spaces
本文研究了张量积空间中的直径二性质与Daugavet性质，基于弱算子Daugavet性质（WODP）推导出一系列新的充分条件，改进了现有文献结果并推进了相关公开问题的解决。
A Banach space $X$ is said to have the Daugavet property if every operator $T: X\to X$ of rank~$1$ satisfies $\|Id+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy of $\ell_{1}$. However, $X$ need not contain a copy of $L_{1}$. We also study pairs of spaces $X\subset Y$ and operators $T: X\to Y$ satisfying $\|J+T\|=1+\|T\|$, where $J: X\to Y$ is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we…
We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative.
A Banach space $X$ has the Daugavet property if the Daugavet equation $\|\Id + T\|= 1 + \|T\|$ holds for every rank-one operator $T:X \longrightarrow X$. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like $\|g(T)\|=f(\|T\|)$ for some functions $f$ and $g$) lead in fact to the Daugavet property of the space. On the other hand there are equations (for example $\|\Id + T\|= \|\Id - T\|$) that lead to new, strictly weaker properties of Banach spaces.
The Operator Daugavet Property in Semifinite Noncommutative $L_1$-Spaces
本文证明扩散半有限 von Neumann 代数上的 $L_1$ 空间与任意非零 Banach 空间的射影张量积具有算子 Daugavet 性质，且见证算子可为压缩算子，推广了经典结果。
The Daugavet property in symmetric operator spaces
本文刻画了附属于半有限无原子von Neumann代数且具有Daugavet性质的对称算子空间，其为$L_1(\u039c,\u03c4)$或$\u039c$，相关结果扩展了已有实对称函数空间的Daugavet性质研究。
The Daugavet equation for polynomials on C$^*$-algebras and JB$^*$-triples
该研究证明满足Daugavet性质的JB$^*$-三元系（含C$^*$-代数）均满足更强的多项式Daugavet性质，且交替Daugavet性质也有对应结论，推进了泛函分析中相关方程的研究。
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternati…
Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the Daugavet index of thickness, which helps us solve an open problem which connect the Daugavet indices with the Daugavet equation. Moreover, we will improve the formerly known estimates of the behavior of Daugavet index on direct sums of Banach spaces by establishing sharp bounds. As a consequence of our results we prove that, for every $0<δ<2$, there exists a Bana…
A Banach space $X$ is said to have the alternative Daugavet property if for every (bounded and linear) rank-one operator $T:X\longrightarrow X$ there exists a modulus one scalar $ω$ such that $\|Id + ωT\|= 1 + \|T\|$. We give geometric characterizations of this property in the setting of $C^*$-algebras, $JB^*$-triples and their isometric preduals.
A Banach space $X$ is said to have the Daugavet property if every rank-one operator $T:X\longrightarrow X$ satisfies $\|Id + T\| = 1 + \|T\|$. We give geometric characterizations of this property in the settings of $C^*$-algebras, $JB^*$-triples and their isometric preduals. We also show that, in these settings, the Daugavet property passes to ultrapowers, and thus, it is equivalent to an stronger property called the uniform Daugavet property.
Let $X$ be a rearrangement-invariant space. An operator $T: X\to X$ is called narrow if for each measurable set $A$ and each $ε> 0$ there exists $x \in X$ with $x^2= χ_A, \int x d μ= 0$ and $\| Tx \| < ε$. In particular all compact operators are narrow. We prove that if $X$ is a Lorentz function space $L_{w,p}$ on [0,1] with $p>2$, then there exists a constant $k_X>1$ so that for every narrow projection $P$ on $L_{w,p}$ $\| Id - P \| \geq k_X. $ This generalizes earlier results on $L_p$ and partially answers a question of E. M. Semenov. Moreover we prove that every rearrangement-invariant func…
A group action approach to the Daugavet property
本文引入G-Daugavet性质，统一了经典Daugavet性质与替代Daugavet性质，通过G-切片和闭凸G-不变包给出刻画，并发现群作用可在经典自反空间上产生新行为，与凸传递性、几乎传递性及有限维旋转问题相关。
The aim of this manuscript is to study \emph{spear operators}: bounded linear operators $G$ between Banach spaces $X$ and $Y$ satisfying that for every other bounded linear operator $T:X\longrightarrow Y$ there exists a modulus-one scalar $ω$ such that $$ \|G + ω\,T\|=1+ \|T\|. $$ To this end, we introduce two related properties, one weaker called the alternative Daugavet property (if rank-one operators $T$ satisfy the requirements), and one stronger called lushness, and we develop a complete theory about the relations between these three properties. To do this, the concepts of spear vector an…
This paper contains the following results: a) Suppose that X is a non-trivial Banach space and L is a non-empty locally compact Hausdorff space without any isolated points. Then each linear operator T: C_{0}(L,X)\to C_{0}(L,X), whose range does not contain C_{00} isomorphically, satisfies the Daugavet equality ||I+T||=1+||T||. b) Let Γbe a non-empty set and X, Y be Banach spaces such that X is reflexive and Y does not contain c_{0} isomorphically. Then any continuous linear operator T: c_{0}(Γ,X)\to Y is weakly compact.
We characterise narrow and strong Daugavet operators on $C(K,E)$-spaces; these are in a way the largest sensible classes of operators for which the norm equation $\|Id+T\| = 1+\|T\|$ is valid. For certain separable range spaces $E$ including all finite-dimensional ones and locally uniformly convex ones we show that an unconditionally pointwise convergent sum of narrow operators on $C(K,E)$ is narrow, which implies for instance the known result that these spaces do not have unconditional FDDs. In a different vein, we construct two narrow operators on $C([0,1],\ell_1)$ whose sum is not narrow.
Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections
研究有限余维子空间的投影常数与最小投影，证明超平面投影常数为2，并探讨C[0,1]空间中投影常数的范围。
We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is octahedral for every non-zero Banach space $Y$. They are those satisfying that, for every finite dimensional subspace $Z$, $\ell_\infty$ can be finitely-representable in a part of $X$ kind of $\ell_1$-orthogonal to $Z$. We also prove that $L(Y,X)$ is octahedral for every $Y$ if, and only if, $L(\ell_p^n,X)$ is octahedral for every $n\in\mathbb N$ and $1<p<\infty$. Finally, we find examples of Banach spaces satisfying the above conditions like $\Lip(M)$ spaces with octahedral norms or $L_1$-preduals with the Daugavet propert…
Slicely countably determined points in Banach spaces
本文定义并系统研究 Banach 空间有界凸集的 SCD 点，刻画其在 $L_1$-预对偶、直和、张量积及 Lipschitz-free 空间中的性质，并用于 Daugavet 性质与方程。
We solve some open problems regarding diameter two properties within the class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw transform. Namely, we show that: the diameter two property, the strong diameter two property, and the symmetric strong diameter two property are all different for these spaces of Lipschitz functions; the space $\operatorname{Lip}_0(K_n)$ has the symmetric strong diameter two property for every $n\in \mathbb{N}$, including the case of $n=2$; every local norm-one Lipschitz function is a Daugavet point.
Generating operators between Banach spaces
本文定义 Banach 空间间的生成算子，刻画其伴随与和空间行为，并研究其范数可达性及闭凸包生成非扩张算子的条件。
In this paper the boundary value problem for one class of the operator-differential equations of the third order on a semi-axis, where one of the boundary conditions is perturbed by some linear operator is researched. There are received sufficient conditions on the operator coefficients of the considered boundary value problem, providing its correct and univalent resolvability in Sobolev type space.
In the present paper, we consider the solvability of positive solutions of nonlinear integral equations by means of investigating non-linear Markov operators. To solve the problem we find necessary and sufficient condition for the surjectivity of nonlinear Markov operators.
Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit is the minimal-norm solution to the equation $B(u)=f$. A global convergence theorem is proved for DSM for equation (*) with monotone $C_{l…
Let $n\in\mathbb{N}$ and let $A$ be a closed linear operator (everywhere bounded or unbounded). In this paper, we study (among others) equations of the type $A^*A=A^n$ where $n\geq2$ and see when they yield $A=A^*$ (or a weaker class of operators). In case $n\geq3$, we have in fact a new class of operators which could placed right after orthogonal projections and just before normal operators.
In this paper, applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by M.S. Livšic and Y. Avishai, are considered. An approach to the inverse scattering problem and to the wave equations is presented, based on the Livšic operator colligation theory (or vessel theory) in the case of commuting bounded nonselfadjoint operators in a Hilbert space, when one of the operators belongs to a larger class of nondissipative operators with asymptotics of the corresponding nondissipative curves. The generalized Gelfand-Levitan-Marchenko equa…
In \cite{Positive semidefinite solutions}, T. Furuta discusses the existence of positive semidefinite solutions of the operator equation $\sum_{j=1}^{n}A^{n-j}XA^{j-1}=B$. In this paper, we shall apply Grand Furuta inequality to study the operator equation. A generalized special type of $B$ is obtained due to \cite{Positive semidefinite solutions}.
On the operator equation AX-XB=C
本文探讨量子理论问题如何促进纯数学中矩阵与算子理论的研究，核心是求解算子方程 AX-XB=C，并应用其解处理广义方程及猴痘疾病模型。
Let $A$ be a densely defined closed operator in a complex Banach space $X.$ Conditions for left invertibility of operators of the form $\sum_{j=1}^\infty a_j (α_j -A)^{-1}$ are given. Several examples are considered.
We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for Cesàro bounded operators of fractional order. The results obtained fairly generalize the corresponding ones for power-bounded operators. In passing, we give an extension of the mean ergodic theorem. Examples are given to illustrate the theory.
In this paper, we study the existence of solutions of some kinds of operator equations via operator inequalities. First, we investigate characterizations of operator order $A\geqslant B >0$ and chaotic operator order log $A \geqslant$ log $B$ for positive definite operators $A$, $B$ in terms of operator equations, and generalize the results in \cite{CSLin}. Then, we introduce applications of complete form of Furuta inequality in operator equations. Some kinds of operator equations are researched and related characterizations of solutions are proved.
The second part of our work on operator synthesis deals with individual operator synthesis of elements in some tensor products, in particular in Varopoulos algebras, and its connection with linear operator equations. Using a developed technique of ``approximate inverse intertwining'' we obtain some generalizations of the Fuglede and the Fuglede-Weiss theorems. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations.
Abstract Volterra integral equations of wave type with almost sectorial operators
本研究针对带几乎扇形算子的波动型抽象Volterra积分方程，借助几乎扇形算子的泛函演算构造包含其传播子的有界线性算子族，探究方程经典解的存在性、唯一性及显式解算子，并给出该算子族的相关性质。
In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method is also presented for stable solutions.
In this note we present a framework which allows to prove an abstract existence result for evolution equations with pseudo-monotone operators. The assumptions on the spaces and the operators can be easily verified in concrete examples.
We study weak and strong solutions of nonlinear non-compact operator equations in abstract spaces of adapted random points. The main result of the paper is similar to Schauder's fixed-point theorem for compact operators. The illustrative examples explain how this analysis can be applied to stochastic differential equations.
Let $K$ be an absolutely convex infinite-dimensional compact in a Banach space $\mathcal{X}$. The set of all bounded linear operators $T$ on $\mathcal{X}$ satisfying $TK\supset K$ is denoted by $G(K)$. Our starting point is the study of the closure $WG(K)$ of $G(K)$ in the weak operator topology. We prove that $WG(K)$ contains the algebra of all operators leaving $\overline{\lin(K)}$ invariant. More precise results are obtained in terms of the Kolmogorov $n$-widths of the compact $K$. The obtained results are used in the study of operator ranges and operator equations.
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily apply this Laplace transform to n+1 dimensional partial differential equations. Further, all the analytic solutions to an initial value problem of an arbitrary order linear partial differential equation are expressed in these abstract operators. By writing abstract operators in this class into integral forms, the solutions in operator form are represented i…
We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the entry A is separated from the spectrum of C, and D is a bounded operator, we obtain a representation for the strong solution X to the Sylvester equation XA-CX=D in the form of an operator Stieltjes integral…
In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on Banach space with locally monotone operators, which is a generalization of the classical result by J.L. Lions for monotone operators. In particular, we show that local monotonicity implies the pseudo-monotonicity. The main result is applied to various types of PDE such as reaction-diffusion equations, generalized Burgers equation, Navier-Stokes equation, 3D Leray-$α$ model and $p$-Laplace equation with non-monotone perturbations.
In this paper, the existence and uniqueness of solution of the Cauchy problem for abstract Boussinesq equation is obtained. By applying this result, the Cauchy problem for systems of Boussinesq equations of finite or infinite orders are studied
Iterative solution of linear equations with unbounded operators
本文针对希尔伯特空间中的可解线性方程，构造了一种收敛的迭代过程。
We obtain estimate of the exponential decay rate of semigroup associated with second order linear differential equation $u"+Du'+Au=0$ in Hilbert space. We assume that $A$ is a selfadjoint positive definite operator, $D$ is an accretive sectorial operator and $\Ree D\geqδA$, $δ>0$. We obtain a location of the spectrum of a pencil associated with linear differential equation.
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