Quentin Menet
Let $(X,(p_j))$ be a Fréchet space with a Schauder basis and without continuous norm, where $(p_j)$ is an increasing sequence of seminorms inducing the topology of $X$. We show that $X$ satisfies the Invariant Subspace Property if and only if there exists $j_0\ge 1$ such that $\ker p_{j+1}$ is of finite codimension in $\ker p_{j}$ for every $j\ge j_0$.
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
A Fréchet space $X$ satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace $M$ in $X$, each continuous operator on $M$ possesses a non-trivial invariant subspace (resp. subset). In this paper, we exhibit a family of non-normable separable infinite-dimensional Fréchet spaces satisfying the Hereditary Invariant Subspace Property and we show that many non-normable Fréchet spaces do not satisfy this property. We also state sufficient conditions for the existence of a continuous operator without non-trivial invariant subset and deduce a…
Understanding the complemented subspaces of $L_p$ has been an interesting topic of research in Banach space theory since 1960. 1999, Alspach proposed a systematic approach to classifying the subspaces of $L_p$ by introducing a norm given by partitions and weights. This paper shows that with Alspach Norm we are able to classify some complemented subspaces of $L_p, 2<p<\infty$.
We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.
We develop the notion of $g$-angle between two subspaces of a normed space. In particular, we discuss the $g$-angle between a $1$-dimensional subspace and a $t$-dimensional subspace for $t\geq 1$ and the $g$-angle between a $2$-dimensional subspace and a $t$-dimensional subspace for $t\geq 2$. Moreover, we present an explicit formula for the $g$-angle between two subspaces of $\ell^{p}$ spaces.
Birkhoff-James orthogonality in certain tensor products of Banach spaces II
本文研究Banach空间张量积中初等张量的Birkhoff-James正交性，通过刻画Lebesgue-Bochner空间中元素的正交性，建立了张量正交性与分量元素正交性之间的关系。
The main result: the dual of separable Banach space $X$ contains a total subspace which is not norming over any infinite dimensional subspace of $X$ if and only if $X$ has a nonquasireflexive quotient space with the strictly singular quotient mapping.
In this note we give a simple proof that every subspace of L_p, 2<p<infinity, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights.
We introduce a new $2$-norm on a normed space using a semi-inner product $g$ on the space. Using the $2$-norm, we propose a formula for the $g$-angle between $2$-dimensional subspaces in the space. Our formula serves as a revision of the one proposed by Nur {\it et al.} \cite{Nur1}.
We show that, if $X$ is a closed subspace of a Banach space $E$ and $Z$ is a closed subspace of $E^*$ such that $Z$ is norming for $X$ and $X$ is total over $Z$ (as well as $X$ is norming for $Z$ and $Z$ is total over $X$), then $X$ and the pre-annihilator of $Z$ are complemented in $E$ whenever $Z$ is $w^*$-closed or $X$ is reflexive.
In the first part of our paper, we show that $\ell_\infty$ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as $\ell_1(\mathfrak{c})$, also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of $c_0(ω_1)$ admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a $C^{\infty}$-smooth norm. Finally, we …
If a separable Banach space $X$ is such that for some nonquasireflexive Banach space $Y$ there exists a surjective strictly singular operator $T:X\to Y$ then for every countable ordinal $α$ the dual of $X$ contains a subspace whose weak$^*$ sequential closures of orders less than $α$ are not norming over any infinite-dimensional subspace of $X$ and whose weak$^*$ sequential closure of order $α+1$ coincides with $X^*$
Necessary and sufficient conditions for Banach space to be(isometrically isomorphic to) a dual space will be given.
On subspaces whose weak* derived sets are proper and norm dense
本文研究 Banach 空间对偶中弱*导出集的长链结构，将 Ostrovskii 的结果推广到任意非拟自反且含可分对偶无穷维子空间的 Banach 空间，证明对任意可数后继序数都存在弱*导出集为真子集且范数稠密的子空间。
We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but no subspace of higher dimension. This gives a $n$-by-$n$ version of the solutions given by Read and Rmoutil to problems of Singer and Godefroy. We also study the existence of strongly proximinal subspaces of finite codimension, showing that for every $1<n<\infty$ and $1\leqslant k <n$, there is a Banach space…
On phase and norm retrieval by subspaces
本文通过子空间研究相位和范数检索，证明了超平面允许范数检索的条件，并分类了能进行范数检索的子空间，同时提供了相关基本结果的替代证明。
G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this property. This is done through a study of the relation between the following two sentences where X is a Banach space and Y is a closed subspace of finite codimension in X: (A) Y is proximinal in X. (B) The annihilator of Y consists of norm-attaining functionals. We prove that these are equivalent if X is the Read's space. Moreover, we prove that any non-reflexi…
Quasi-greedy Markushevich bases, duality and norming subspaces
本文证明拟贪婪 Markushevich 基的对偶基张成赋范子空间，并将结果推广至较弱形式及文献中的其他类贪婪性质。
Many of the known complemented subspaces of L_p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well-known complemented subspaces of L_p. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of L_p. Using this we define a space Y_n with norm…
Failure of the wandering subspace property for analytic norm-increasing $3$-isometries
本文通过在价为 2 的无根拟 Brown 有向树上构造不具游荡子空间性质的解析范数递增 3-等距加权移位，否定回答了 Shimorin 的问题。
The wandering subspace problem for an analytic norm-increasing $m$-isometry $T$ on a Hilbert space $\mathcal H$ asks whether every $T$-invariant subspace of $\mathcal H$ can be generated by a wandering subspace. An affirmative solution to this problem for $m=1$ is ascribed to Beurling-Lax-Halmos, while that for $m=2$ is due to Richter. In this paper, we capitalize on the idea of weighted shift on one-circuit directed graph to construct a family of analytic cyclic $3$-isometries, which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a o…
We study the class of Banach spaces $X$ such that the locally convex space $(X,μ(X,Y))$ is complete for every norming and norm-closed subspace $Y \subset X^*$, where $μ(X,Y)$ denotes the Mackey topology on $X$ associated to the dual pair $\langle X,Y\rangle$. Such Banach spaces are called fully Mackey complete. We show that fully Mackey completeness is implied by Efremov's property ($\mathcal{E}$) and, on the other hand, it prevents the existence of subspaces isomorphic to $\ell_1(ω_1)$. This extends previous results by Guirao, Montesinos and Zizler [J. Math. Anal. Appl. 445 (2017), 944-952] a…
Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges
本文证明了 $\ell_p(Γ)$ 的稠密子空间 $\mathcal{Y}_p$ 具有局部有限依赖的 $C^{\infty}$-光滑范数，且可逼近 $p$-范数，并讨论了其包含稠密算子值域的条件及不可分算子值域的光滑性限制。
The variance conjecture in Asymptotic Convex Geometry stipulates that the Euclidean norm of a random vector uniformly distributed in a (properly normalised) high-dimensional convex body $K\subset {\mathbb R}^n$ satisfies a Poincaré-type inequality, implying that its variance is much smaller than its expectation. We settle the conjecture for the cases when $K$ is the unit ball of the operator norm in classical subspaces of square matrices, which include the subspaces of self-adjoint matrices. Through the estimates we establish, we are also able to show that the unit ball of the operator norm in…
The Stability and Density of Norm Attaining Operators on Reducing Subspaces
本文研究无限维希尔伯特空间上约化子空间范数达到算子的类$\beta(H)$，证明其在算子范数下稠密，并引入子类$\beta_0(H)$，刻画其有限秩与紧扰动稳定性，阐明谱结构。
We prove the simmetry of the Burkill-Cesari integral and discuss its continuity with respect to both the $ BV$ norm of Aumann and Shapley and to the Lipschitz norm. As a consequence, we provide an existence result of a value, different from the Aumann and Shapley's one, on a subspace of $AC_\infty$.
A subspace $H$ of a rearrangement invariant space $X$ on $[0,1]$ is strongly embedded in $X$ if, in $H$, convergence in $X$-norm is equivalent to convergence in measure. We obtain necessary and sufficient conditions on an Orlicz function $M$, under which the unit ball of an arbitrary strongly embedded subspace in the Orlicz space $L_M$ has equi-absolutely continuous norms in $L_M$.
About smooth and non-poor subspaces of Daugavet spaces
研究非完备范数空间中具有Daugavet性质的光滑范数，以及其对偶范数的非光滑性，并探讨Quasilacunary Müntz空间作为C[0,1]的子空间的性质。
In this note we prove that $\frac{1}{n!} \sum_π (\sum_{i=1}^n |x_i a_{i,π(i)} |^2)^{1/2}$ is equivalent to a Musielak-Orlicz norm $\norm{x}_{\sum M_i}$. We also obtain the inverse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak-Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak-Orlicz spaces into L_1.
On Completions and Dense Subspaces of Strongly Facially Symmetric Spaces
本文研究强面部对称空间的完备化与稠密子空间的性质，引入面稠密条件保证其完备化后仍为强面部对称空间，证明中性SFS-空间的不变稠密子空间可继承其结构，并给出相关例子与反例。
For Fr{é}chet spaces E and F we write (E,F) \in {B} if every continuous linear operator from E to F is bounded. Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We obtain a necessary and sufficient condition for (E,F) \in {B} when F = λ^{l}(B). We say that a triple (E,F,G) has the bounded factorization property and write (E,F,G) \in {BF} if each continuous linear operator T : E \longrightarrow G that factors over F is bounded. We extend some results in \cite{Ter03} to l-Köthe spaces and obtain a sufficient condition for (E,λ…
This paper deals with study of Birkhoff-James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For arbitrary Hilbert space $ \mathbb{H},$ we also study orthogonality to subspace of the space of linear operators $L(\mathbb{H}), $ both with respect to operator norm as well as numerical radius norm.
We show that, for vector spaces in which distance measurement is performed using a gauge, the existence of best coapproximations in $1$-codimensional closed linear subspaces implies in dimensions $\geq 2$ that the gauge is a norm, and in dimensions $\geq 3$ that the gauge is even a Hilbert space norm. We also show that coproximinality of all closed subspaces of a fixed dimension implies coproximinality of all subspaces of all lower finite dimensions.
Interpolation of subspaces of infinite codimension
本文研究无限余维闭子空间替换后插值空间的比较问题，提出商 J 范数作为处理多种相关情形的有效工具。
We study when diameter two properties pass down to subspaces. We obtain that the slice two property (respectively diameter two property, strong diameter two property) passes down from a Banach space $X$ to a subspace $Y$ whenever $Y$ is complemented by a norm one projection with finite-dimensional kernel (respectively the quotient $X/Y$ is finite dimensional, $X/Y$ is strongly regular). Also we study the same problem for dual properties of the above ones, as having octahedral, weakly octahedral or 2-rough norm.
Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm.
G-strong subdifferentiability and applications to norm attaining subspaces
本文在群不变映射框架下研究自反性与强次可微性，证明对偶范数G-强次可微蕴含Banach空间G-自反，并推广多个经典泛函分析定理，最终给出范数可达泛函集合中存在闭Banach子空间的充分条件。
Essential norms of composition operators and multipliers acting between different Hardy spaces
本文计算了从$L^p$空间闭子空间到$L^q$空间子空间的包含、复合及乘子算子的本性范数，条件为$p>q$。
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 $L_1(μ)$.
We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $\ell_p(X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space $\ell…
In this note we study Banach spaces of traces of real polynomials on $\mathbb R^n$ to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.
Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces $Y$, as well as the inverse question, that is, the possible sizes of the metric space $M$ given that such a subspace $Y$ exists. We also show that if the metric space $M$ is $σ$-precompact, then th…
We show that for any bounded operator $T$ acting on an infinite dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^*$ does not consist entirely of eigenvalues, we can find such rank one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite rank perturbations of arbitrarily small norm, but not necessarily of rank one.
We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or …
Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the Gâteaux and Fréchet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_δ$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ a…
We give an intrinsic characterisation of the separable reflexive Banach spaces that embed into separable reflexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht.
In this paper, a sufficient condition for the existence of hyperinvariant subspace of compact perturbations of multiplication operators on some Banach spaces is presented. An interpretation of this result for compact perturbations of normal and diagonal operators on Hilbert space is also discussed. An improvement of a result of [FX12] for compact perturbations of diagonal operators is also obtained.
Given an $N$-dimensional subspace $X$ of $L_p([0,1])$, we consider the problem of choosing $M$-sampling points which may be used to discretely approximate the $L_p$ norm on the subspace. We are particularly interested in knowing when the number of sampling points $M$ can be chosen on the order of the dimension $N$. For the case $p=2$ it is known that $M$ may always be chosen on the order of $N$ as long as the subspace $X$ satisfies a natural $L_\infty$ bound, and for the case $p=\infty$ there are examples where $M$ may not be chosen on the order of $N$. We show for all $1\leq p<2$ that there e…
We construct a continuous linear operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an operator acting on a Fréchet space without a continuous norm. The construction is based on the ideas due to C. Read who constructed a continuous operator without non-trivial invariant subspaces on the Banach space $\ell_1$.
For a subspace $X$ of functions from $L_2$ we consider the minimal number $m$ of nodes necessary for the exact discretization of the $L_2$-norm of the functions in $X$. We construct a subspace such that for any exact discretization with $m$ nodes there is at least one negative weight.
It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators in a real Hilbert space.
It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators on a real Hilbert space.
We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small," contains a further subspace isometric to V. This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of convex bodies) and allows to solve several problems stated in the 1980s by V. Milman. The proofs are probabilistic and depend on careful analysis of images of convex sets under Gaussian linear maps.
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(μ)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Gu$\acute{e}$don and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. …
We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read and Rmoutil. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $NA(X)$ for the original norm is not "too large". The construction can be applied to every Banach space containing $c_0$ and having a countable…
We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.
Polyhedral norms and smooth Hahn-Banach extension
本文在多面体范数下，给出了子空间上的光滑泛函存在保范光滑延拓到全空间的充要条件，即该光滑泛函的唯一绝对范数达到点是子空间单位球和全空间单位球的极点，并证明了该结果在非多面体范数下不成立。
The \emph{relative projection constant} $λ(Y, X)$ of normed spaces $Y \subset X$ is defined as $λ(Y, X) = \inf \{ ||P|| : P \in \mathcal{P}(X, Y) \}$, where $\mathcal{P}(X, Y)$ denotes the set of all continuous projections from $X$ onto $Y$. By the well-known result of Bohnenblust for every $n$-dimensional normed space $X$ and its subspace $Y$ of codimension $1$ the inequality $λ(Y, X) \leq 2 - \frac{2}{n}$ holds. The main goal of the paper is to study the equality case in the theorem of Bohnenblust. We establish an equivalent condition for the equality $λ(Y, X) = 2 - \frac{2}{n}$ and present …
We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log (dim X) = O(log (dim V)) and (2) every subspace of X, whose dimension is not "too small," contains a further well-complemented subspace nearly isometric to V. This sheds new light on the structure of large subspaces or quotients of normed spaces (resp., large sections or linear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by V. Milman. The pr…
We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $\varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $\varepsilon$ such that $T+F$ has an invariant subspace of infinite dimension and codimension. A version of this result was proved in \cite{T19} under additional spectral conditions for $T$ or $T^*$. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.
$2$-strong uniqueness of a best approximation and of minimal projections in complex polytope norms and their duals
本文研究了复多面体范数及其对偶空间中最佳逼近的$2$-强唯一性，证明两类空间不相交，并在特定条件下将该性质推广至极小投影，得出三维复空间二维子空间上范数大于$1$的极小投影是$2$-强唯一的。
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。