K. S. Tulenov
We identify symmetric quasi-Banach range of the discrete Calderón operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.
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We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space.
This paper is devoted to the study of reproducing kernel Hilbert spaces. We focus on multipliers of reproducing kernel Banach and Hilbert spaces. In particular we tried to extend this concept and prove some theorems.
In this paper Hilbert spaces are characterized among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical here: If X is a real Banach space isomorphic to a Hilbert space and convex-transitive with respect to the isometric finite-dimensional perturbations of the identity, then X is already isometric to a Hilbert space.
On compact (limited) operators between Hilbert and Banach spaces
本文研究 Hilbert 到 Banach 空间间线性算子集合的紧性，给出集体紧（有限）集的正交刻画。
This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on reproducing kernels in vector-valued reproducing kernel Hilbert spaces. In particular we extend reproducing kernels to relative reproducing kernels and prove some theorems in this subject.
We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell_2$ is the only non ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell_p$, for $1\leq p <2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976.
Uniform Embeddings of Banach Space Spheres into Hilbert Spaces and Property (H)
本文证明可分无穷维实 Banach 空间单位球面均匀嵌入 Hilbert 空间蕴含该空间具有 Kasparov–Yu 性质 (H)。
We give a review of results on the operator-norm convergence of the Trotter product formula on Hilbert and Banach spaces, which is focused on the problem of its convergence rates. Some recent results concerning evolution semigroups are presented in details.
Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property
本文研究了Hilbert和Banach空间中将小球映射为凸集的非线性光滑映射，提出新的温和条件以保证局部凸性，并探讨了Banach空间非线性映射局部凸性相关的开放问题。
We prove that a Hilbert space frame $\fti$ contains a Riesz basis if every subfamily $\ftj , J \subseteq I ,$ is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements.
In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.
We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use $\ell_Ψ$-Hilbertian and $\infty$-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition in a Hilbert space and introduce the class of spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and $M$-cotype of a Banach space and study the properties of unconditional Schaude…
In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $γ$-summing or $γ$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ of pre-Hilbert-Schmidt operators as the class of all operators $u:E\to F$ such that $w\circ u \circ v$ is Hilbert-Schmidt for every bounded operator $v: H_1\to E$ and every bounded operator $w:F\to H_2$, where $H_1$ et $H_2$ are Hilbert spaces. Besides the trivial case where one of the…
Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.
Paley-Wiener theorem for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach spaces. We show that our results give Paley-Wiener theorem for frames for Hilbert spaces.
We give a direct proof of the fact that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it uniformly embeds into a Hilbert space.
We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.
We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.
Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given an arbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affine hyperplanes of $X$ such that every $H\in\mathcal{H}$ passes through some point $x_{H}\in C$, and $C=\{x_H : H\in\mathcal{H}\}$, what conditions are necessary and sufficient for the existence of a $C^{1,1}$ convex hypersurface $S$ in $X$ such that $H$ is tangent to $S$ at $x_H$ for every $H\in\mathcal{H}$? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class $C^{1, ω}$ in Hilb…
We show that if the dyadic Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the Hilbert transform, with a linear relation of the bounds. This result is the counterpart of [arXiv:2212.00090] where the opposite bound was proven.
In this paper we consider series expansions via a frame and a non-frame and the possibilities for interchange of the two sequences, both in the Hilbert and Banach space setting. First we give a characterization of frame-related concepts in Banach spaces (atomic decompositions, Banach frames, $X_d$-Riesz bases, $X_d$-frames, $X_d$-Bessel sequences, and sequences satisfying the lower $X_d$-frame condition). We also determine necessary and sufficient conditions for operators to preserve the type of the concepts listed above. Then we discuss differences and relationships between expansions in a Ba…
Continuing the research on the Banach-Saks and Schur properties started in (cf. M. Frank, A. A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert C*-modules (submitted)) we investigate analogous properties in the module context. As an environment serves the class of Hilbert C*-modules. Some properties of weak module topologies on Hilbert C*-modules are described. Natural module analogues of the classical weak Banach-Saks and the classical Schur properties are defined and studied. A number of useful characterizations of properties of Hilbert C*-modules is obtained. In particular, some i…
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
In this paper we study some basic properties of bicomplex linear operators on bicomplex Hilbert spaces. Further we discuss some applications of Hahn-Banach theorem on bicomplex Banach modules. We also introduce and discuss some bicomplex holomorphic function spaces and prove Littlewood's Subordination principle for bicomplex Hardy space.
Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in…
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. …
In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a certain type of function constituted a Lyapunov function for some time-invariant linear system. Six years later, it was obtained that this property holds if and only if the Banach space enjoys the self-extension property. However, the knowledge of these spaces needed to be extended in order to make useful this characterization, since there were little information on which classic Banach spaces satisfy this property and its relations with other classic properties. We present the self-extension property in a wider frame, r…
Periodicity in Banach algebras
研究Banach代数上等距同构与左乘子复合算子的周期性，给出其周期元稠密的充要条件，并应用于C*-代数上的广义加权移位。
We give a criterion ensuring that the elementary class of a modular Banach space E (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of E) consists of all direct sums E\oplus_m H, where H is an arbitrary Hilbert space and \oplus_m denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.
We show that, for quasi-greedy bases in Hilbert spaces, the associated conditionality constants grow at most as $O(\log N)^{1-ε}$, for some $ε>0$, answering a question by Temlyakov. We show the optimality of this bound with an explicit construction, based on a refinement of the method of Olevskii. This construction leads to other examples of quasi-greedy bases with large $k_N$ in Banach spaces, which are of independent interest.
W.B. Johnson has constructed a series of Banach spaces non isomorphic to the Hilbert one that have the hereditarily approximation property (shortly hereditarily AP): all their subspaces also have the AP. All these examples were ''sufficiently'' non-symmetric and this fact allows Johnson to ask: whether there exists any Banach space $X$ with symmetric (or, at least, subsymmetric) basis, distinct from the Hilbert space such that each its subspace has the AP? In this paper is shown that there is a Banach space with a subsymmetric basis (non-equivalent to any symmetric one), which enjoys the hered…
We construct finitely generated groups with arbitrary prescribed Hilbert space compression αfrom the interval [0,1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. These groups are also the…
Classification of ill-posedness for bounded linear operators in Banach spaces
针对Banach空间中线性算子不适定性定义不统一的问题，梳理现有相关定义，完成I型、II型不适定性分类并讨论边界混合情形与算子实例。
For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em not} identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(Ω)$ and $H^{-1}(Ω)$. In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $\ell^2$-Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorph…
Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces
该研究构造了可分希尔伯特空间中满足特定级数发散条件的非强$(1+\varepsilon_n)$有界Markushevich基，证明条件的紧性，并推广到任意巴拿赫空间，得到经任意置换都不是Schauder基的对应基的存在性结论。
Operator ordering by ill-posedness in Hilbert and Banach spaces
本文提出一种基于有界算子级联的算子不适定性排序方法，证明其与 Mathé 和 Hofmann 定义的等价性，并将其推广至 Banach 空间及非线性问题。
We obtain sharp approximation results for into nearisometries between Lp spaces and nearisometries into a Hilbert space. Our main theorem is the optimal approximation result for nearsurjective nearisometries between general Banach spaces.
We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary represent…
Following Davie's example of a Banach space failing the approximation property [D], we show how to construct a Banach space E which is asymptotically Hilbertian and fails the approximation property. Moreover, the space E is shown to be a subspace of a space with an unconditional basis which is ``almost'' a weak Hilbert space and which can be written as the direct sum of two subspaces all of whose subspaces have the approximation property.
We define the frame potential for a Schauder frame on a finite dimensional Banach space as the square of the $2$-summing norm of the frame operator. As is the case for frames for Hilbert spaces, we prove that the frame potential can be used to characterize finite unit norm tight frames (FUNTFs) for finite dimensional Banach spaces. We prove the existence of FUNTFs for a variety of spaces, and in particular that every $n$-dimensional complex Banach space with a $1$-unconditional basis has a FUNTF of $N$ vectors for every $N\geq n$. However, many interesting results on FUNTFs and sums of rank on…
It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a result of Paulsen and Smith who proved the same for Banach algebras acting on Hilbert spaces.
Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a certain subset of the Banach bidual module V**. We give another (equivalent) definition of quasi-multipliers for Hilbert C*-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C*-algebras, provided these C*-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topol…
In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained can be regarded as partial answers to the well-known Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.
A map f between two metric spaces (X,d_1) and (Y,d_2) is called a coarse embedding of X into Y if there exist two nondecreasing functions phi_1, phi_2:[0,\infty) --> [0,\infty) such that: phi_1(d_1(x,y)) \leq d_2(f(x),f(y)) \leq phi_2(d_1(x,y)) for all x, y in X, and phi_1(t) tends to \infty as t tends to \infty. We characterize those quasi-Banach spaces that have a coarse embedding into a Hilbert space.
EP Banach space operators and EP Banach algebra elements are characterized using different kinds of factorizations. The results obtained generalize well-known characterizations of EP matrices, EP Hilbert space operators and EP $C^*$-algebra elements. Furthermore, new results that hold in these contexts are presented.
We obtain the following characterization of Hilbert spaces. Let $E$ be a Banach space whose unit sphere $S$ has a hyperplane of symmetry. Then $E$ is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group ${\rm Iso}\, E$ of $E$ has a dense orbit in S; b) the identity component $G_0$ of the group ${\rm Iso}\, E$ endowed with the strong operator topology acts topologically irreducible on $E$. Some related results on infinite dimentional Coxeter groups generated by isometric reflexions are given which allow to analyse the structure of isometry groups containin…
Several characterizations of EP and normal Moore-Penrose invertible Banach algebra elements will be considered. The Banach space operator case will be also studied. The results of the present article will extend well known facts obtained in the frames of matrices and Hilbert space operators.
We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.
We introduce a flexible almost isometric version of the almost transitivity property of Banach spaces. With the help of this new notion we generalize to several directions a strong recent rotational characterization of Hilbert spaces due to Randrianantoanina. This characterization is a partial answer to the classical Banach-Mazur rotation problem.
The purpose of this note is to show that, if $\mcB$ is a uniformly convex Banach, then the dual space $\mcB'$ has a "Hilbert space representation" (defined in the paper), that makes $\mcB$ much closer to a Hilbert space then previously suspected. As an application, we prove that, if $\mcB$ also has a Schauder basis (S-basis), then for each $A \in \C[\mcB]$ (the closed and densely defined linear operators), there exists a closed densely defined linear operator $A^* \in \C[\mcB]$ that has all the expected properties of an adjoint. Thus for example, the bounded linear operators, $L[\mcB]$, is a $…
Weighted EP Banach space operators and Banach algebra elements are characterized using different kinds of factorizations. The results presented extend well-known characterizations of (weighted) EP matrices, (weighted) EP Hilbert space operators and (weighted) EP $C^*$-algebra elements.
We prove that if a non-atomic separable Banach lattice in a weak Hilbert space, then it is lattice isomorphic to $L_2(0,1)$.
We prove that a Banach space has the uniform approximation property with proportional growth of the uniformity function iff it is a weak Hilbert space.
In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.
We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing absolutely Cesàro bounded operators on $\ell^p(\mathbb{N})$, $1\le p < \infty$, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement very limited number of known examples (see \cite{Shi} and \cite{AS}). In \ci…
On properties of normal operators and self-adjoint operators on smooth Banach spaces
本文探讨光滑巴拿赫空间上正规和幺正算子的性质，扩展了从希尔伯特空间到巴拿赫空间的经典概念，并研究自伴随算子的范数、最小模、数值半径等性质及谱特性。
Moreau's decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics. In this paper, it is extended to reflexive Banach spaces and in the context of generalized proximity measures. This extension unifies and significantly improves upon existing results.
On some new metric characterisations of Hilbert spaces
本文综述了用 Lipschitz 映射替代线性有界映射后，巴拿赫空间刻画为希尔伯特空间的同构版本新结果，重点关注由内积诱导等价范数的存在性。
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $\mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $\mathcal{H}_1$ which contains $\mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and $C^*$-algebra elements.
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