Gromov hyperbolicity of periodic planar graphs

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.     

Inertial manifolds and normal hyperbolicity

Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold.     

Conditions for the Hyperbolicity of a Linear Third-Order Differential Equation in a One-Dimensional Space

We study the general linear third-order differential equation with constant real coefficients in the case of two independent variables. Necessary and sufficient coefficient conditions for the hyperbolicity and strict hyperbolicity of the equation, as well as for the representability of the differential operator occurring in the equation in the form of a composition of first-order operators, are obtained.     

On regular linear relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.     

Outer preserving linear operators

A natural question about linear operators on the Hilbert-Hardy space is answered, motivated by work in geophysical imaging. Namely, which bounded linear operators on the Hardy space preserve the set of all shifted outer functions? A complete characterization is determined, which allows an explicit construction of all such operators. Every operator that preserves the set of shifted outer functions is necessarily a product-composition operator, consisting of composition with a shifted outer function followed by multiplication with a (possibly different) shifted outer function. Such operators represent important physical processes, including the propagation of seismic wave energy through the earth. Applications to seismic imaging are briefly discussed.     

On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces

The main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. We provide a new stability characterization of oblique projection generalized inverses of closed linear operators under T-bounded perturbations, which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space.     

Norm-parallelism and the Davis–Wielandt radius of Hilbert space operators

ABSTRACT

We present a necessary and sufficient condition for the norm-parallelism of bounded linear operators on a Hilbert space. We also give a characterization of the Birkhoff–James orthogonality for Hilbert space operators. Moreover, we discuss the connection between norm-parallelism to the identity operator and an equality condition for the Davis–Wielandt radius. Some other related results are also discussed.     

保不定半正交性的可加映射

设Hi是实或复数域上无限维完备的不定度规空间,Ai是B(Hi)中由单位元I和一个理想生成的子代数,其中B(Hi)表示Hi上所有有界线性算子构成的代数,i=1,2.本文刻画了从A1到A2上双边保不定半正交性的可加满射Ф,即对任意T,S∈A1,T S=0(=)Ф(T) Ф(S)=0.主要结果表明,这样的Ф具有形式Ф(T)=UTV对任意的T∈A1成立,这里U,V是有界线性或共轭线性可逆算子且U U=cI,c是非零实数.     

Nonuniform exponential dichotomies and Lyapunov functions

For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.     

Closed Semistable Operators and Singular Differential Equations

We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which 0 is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of C ₀-semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly perturbed differential equation in a Banach space.     

The sets of monomorphisms and of almost open operators between locally convex spaces

If the set of monomorphisms between locally convex spaces is not empty, then it is an open subset of the space of all continuous and linear operators endowed with the topology of the uniform convergence on the bounded sets if and only if the domain space is normable. The corresponding characterization for the set of almost open operators is also obtained; it is related to the lifting of bounded sets and to the quasinormability of the domain space. Other properties and examples are analyzed.

     

Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

In this article we prove comparative results on the Gromov hyperbolicity of plane domains equipped with the quasihyperbolic metric. By a comparative result we mean one which assumes hyperbolicity in one domain and obtains it in a different domain somehow related to the original one. We derive a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a plane domain Ω* obtained by deleting from the original domain Ω any uniformly separated union of compact sets. We present as well a result about stability of hyperbolicity.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

B(X)上保持约当积的固定点的满射

设B(X)是维数大于等于3的复Banach空间X上有界线性算子全体构成的代数.设A∈B(X),若Ax=x,则称x∈X是算子A的固定点.Fix(A)表示A的所有固定点的集合.本文刻画了B(X)上保持算子的Jordan积的固定点的满射.     

The moduli space of hyperbolic compact complex spaces

In this paper, we construct the moduli space of reduced hyperbolic compact complex spaces. First, we prove an infinitesimal characterization of hyperbolicity using a family of Kobayashi–Royden pseudo-metrics introduced by Venturini and as a consequence we conclude that the property of Landau holds for complex spaces. Finally, we establish this moduli space in the case of locally trivial deformations, and in a more general situation, the case of equisingular deformations.     

Resolvent Tests for Similarity to a Normal Operator

The main result of this paper is the resolvent similarity criterionwhich says that linear growth of the resolvent towards the spectrumis sufficient for a Hilbert space contraction with finite rankdefect operators and spectrum not covering the unit disc tobe similar to a normal operator. Similar results are provedfor operators having a spectral set bounded by a Dini-smoothJordan curve; in particular, a dissipative operator with finiterank imaginary part is similar to a normal operator if and onlyif its resolvent grows linearly towards the spectrum. Relevantresults on the insufficiency of linear resolvent growth notaccompanied by smallness of defect operators are presented.Also it is proved that there is no restriction on the spectrum,other than finiteness, which together with linear resolventgrowth implies similarity to a normal operator. The constructionof corresponding examples depends on a characterization of well-knownAhlfors curves as curves of linear length growth with respectto linear fractional transformations. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

A Gleason–Kahane–Żelazko theorem for the Dirichlet space

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions.We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.     

Gromov hyperbolic spaces and the sharp isoperimetric constant

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.     

Gromov hyperbolicity through decomposition of metric spaces

We study the hyperbolicity of metric spaces in the Gromov sense. We deduce the hyperbolicity of a space from the hyperbolicity of its “building block components”. These results are valuable since they simplify notably the topology of the space and allow to obtain global results from local information. We also study how the punctures and the decomposition of a Riemann surface in Y-pieces and funnels affect the hyperbolicity of the surface.