George Androulakis, Per Enflo
We prove that if T is a strictly singular 1-1 operator defined on an infinite dimensional Banach space X, then for every infinite dimensional subspace Y of X there exists an infinite dimensional subspace Z of Y such that Z contains orbits of T of every finite length and the restriction of T on Z is a compact operator.
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We investigate possible quantifications of strictly singular operators, $l_{p}$-strictly singular operators, $c_{0}$-strictly singular operators, strictly cosingular operators, $l_{p}$-strictly cosingular operators. We prove quantitative, even strengthening versions of well-known results about relationships of these five classes of operators and compact, weakly compact, unconditionally converging operators.
The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of $ω_1$ subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.
We construct a strictly singular non-compact operator on Gowers' and Maurey's space $GM$.
Given $0<p,q, r<\infty $ and $ q<r\le \infty$ we consider the natural embedding $\ell_{p,q}\hookrightarrow \ell_{p,r}$ between Lorenz sequence spaces. We prove that this non-compact embedding is always strictly singular but not finitely strictly singular.
We study the interpolation and extrapolation properties of strictly singular operators between different $L_p$ spaces. To this end, the structure of strictly singular non-compact operators between $L_p-L_q$ spaces is analyzed. Among other things, we clarify the relation between strict singularity and the $L$-characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators.
Based on the recently introduced uniform $λ-$adjustment for closed subspaces of Banach spaces we extend the concept of the strictly singular and finitely strictly singular operators to the sequences of closed subspaces and operators in Banach spaces and prove theorems about lower semi--Fredholm stability. We also state some new open questions related to strict singularity and the geometry of Banach spaces.
Given $1< p < q < \infty$ it is well know that the natural embedding of Lebesgue sequence spaces $\ell_p \hookrightarrow \ell_q$ is strictly singular. In this paper we extend this classical results and show that even the natural non-compact embedding between Lebesgue and weak Lebesgue sequence spaces $\ell_{p}\hookrightarrow \ell_{p,\infty}$ is strictly singular.
Exploiting several $\ell_p$-factorization results for strictly singular operators, we study the strict singularity of the multiplication operator $L_A R_B\colon T\mapsto ATB$ on $\mathcal L(X)$ for various Banach spaces~$X$.
A continuous linear operator $T:E \to F$ is called strictly singular if it cannot be invertible on any infinite dimensional closed subspace of its domain. In this note we discuss sufficient conditions and consequences of the phenomenon $LB(E,F)=L_s(E,F)$, which means that every continuous linear bounded operator defined on $E$ into $F$ is strictly singular.
We present a method for constructing bounded strictly singular non-compact operators on mixed Tsirelson spaces defined either by the families (A_n) or (S_n) of a certain class, as well as on spaces built on them, including hereditarily indecomposable spaces.
Using the notion of $S_ξ$-strictly singular operator introduced by Androulakis, Dodos, Sirotkin and Troitsky, we define an ordinal index on the subspace of strictly singular operators between two separable Banach spaces. In our main result, we provide a sufficient condition implying that this index is bounded by $ω_1$. In particular, we apply this result to study operators on totally incomparable spaces, hereditarily indecomposable spaces and spaces with few operators.
We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces $T_M[(\mc{S}_n,θ_n)]$ satisfying some regularity conditions and present results on existence of strictly singular non-compact operators on subspaces of mixed Tsirelson spaces defined by the families $(\mc{A}_n)_n$ and $(\mc{S}_n)_n$.
We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic $\ell_p$ spaces, $1\leq p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic $\ell_p$ HI spaces.
We construct a Hereditarily Indecomposable Banach space $\eqs_d$ with a Schauder basis \seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\mc{L}_{\diag}(\eqs_d)$ of diagonal operators with respect to the basis \seq{e}{n} contains an isomorphic copy of $\ell_{\infty}(\N)$.
Disjointly strictly singular inclusions between variable Lebesgue spaces
研究有限与无限测度下可变Lebesgue空间包含的不相交严格奇异性，给出基于指数的判据，并证明其与$L$-弱紧性等价。
It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
We study the structure of strictly singular non-compact operators between $L_p$ spaces. Answering a question raised in [Adv. Math. 316 (2017), 667-690], it is shown that there exist operators $T$, for which the set of points $(\frac1p,\frac1q)\in(0,1)\times (0,1)$ such that $T:L_p\rightarrow L_q$ is strictly singular but not compact contains a line segment in the triangle $\{(\frac1p,\frac1q):1<p<q<\infty\}$ of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and st…
V. D. Milman proved in \cite{Milman:70} that the product of two strictly singular operators on $L_p[0,1]$ ($1\le p<\infty$) or on $C[0,1]$ is compact. In this note we utilize Schreier families $§_ξ$ in order to define the class of $§_ξ$-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of $§_ξ$-hereditarily indecomposable Banach spaces and we examine …
Strictly convex norms and the local diameter two property
本文在实函数实心巴拿赫格中引入范数严格单调性以阻止局部直径二性质，证明$c_0(Γ)$上严格凸1-对称范数具此性质，并构造不可数$Γ$的$c_0(Γ)$上带1-无条件基且具局部直径二性质的严格凸范数。
A classification of weakly compact multiplication operators on L(L_p), $1<p<\infty$, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of $\ell_p$-strictly singular operators, and we also investigate the structure of general $\ell_p$-strictly singular operators on L_p. The main result is that if an operator T on L_p, 1<p<2, is $\ell_p$-strictly singular and T_{|X} is an isomorphism for some subspace X of L_p, then X embeds into L_r for all r<2, but X need not be isomorphic to a Hilbert space. It is also shown that if T is convolution…
Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces
研究严格奇异算子在特定Banach空间直和上的紧性性质，证明商代数的nilpotent性及复合算子的紧性特征。
For each $n \in \mathbb{N}$ a Banach space $\mathfrak{X}_{0,1}^n$ is constructed is having the property that every normalized weakly null sequence generates either a $c_0$ or $\ell_1$ spreading models and every infinite dimensional subspace has weakly null sequences generating both $c_0$ and $\ell_1$ spreading models. The space $\mathfrak{X}_{0,1}^n$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of $\mathfrak{X}_{0,1}^n$, for every $S_1,S_2,\ldots,S_{n+1}$ strictly singular operators on $Y$, the operator $S_1S_2\cdots S_{n+1}$ is compact. Moreover, for every subsp…
A strictly convex and smooth Banach space with the Daugavet property
本文在$L_1[0,1]$上构造了接近通常范数的等价范数，使空间严格凸且光滑，且对偶范数具有Daugavet性质，解决了长期未解问题。
Let $X$ and $Y$ be separable Banach spaces and denote by $\sss\sss(X,Y)$ the subset of $\llll(X,Y)$ consisting of all strictly singular operators. We study various ordinal ranks on the set $\sss\sss(X,Y)$. Our main results are summarized as follows. Firstly, we define a new rank $\rs$ on $\sss\sss(X,Y)$. We show that $\rs$ is a co-analytic rank and that dominates the rank $\varrho$ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math., 169 (2009), 221-250]. Secondly, for every $1\leq p<+\infty$ we construct a Banach space $Y_p$ with an unconditional basis such that $\sss\sss…
In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $Δ$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $Γ$ with $\vert Γ\vert \geq 2$, the Banach space $\bigoplus_{γ\in Γ}^{c_0} X_γ$ satisfies the Mazur--Ulam property, whenever the Banach space $X_γ$ is strictly convex with dim$((X_γ)_{\mathbb{R}})\geq 2$ for every $γ$. Moreover we prove that the Banach space $C_0(K,X)$ satisfi…
Strictly singular operators on the Baernstein and Schreier spaces
该文证明在 Baernstein 与 Schreier 空间上，两个严格奇异算子复合为紧算子，并刻画了次对称基的等价性及非补块序列的存在。
A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly midpoint locally uniformly rotund and has the D$2$P. We also prove that for Banach spaces admitting a norm-one finite-co-dimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.
Reflexive and strictly convex Banach spaces without Property (H)
本文证明一大类自反严格凸 Banach 空间（包括特定 $\ell^s$ 和的 $L^{p_j}$ 空间）不满足性质 (H)，推广了该性质的失效情形。
A condition for the presence of a "gap" between symmetric spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. This condition is stated in terms of fundamental functions of spaces and is exact in a certain sense. In parallel, necessary and sufficient conditions for an inclusion of Lorentz spaces to be disjointly strictly singular are obtained.
We construct a family $(\mathcal{X}_\al)_{\al\le ω_1}$ of reflexive Banach spaces with long transfinite bases but with no unconditional basic sequences. In our spaces $\mathcal{X}_\al$ every bounded operator $T$ is split into its diagonal part $D_T$ and its strictly singular part $S_T$.
We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of disjointly strictly singular operators, those not invertible on the span of any disjoint sequence, coincides with that of lattice strictly singular operators, i.e. those not invertible on any (infinite dimensional) sublattice. New results are given which help to clarify the existing relation between these two classes.
We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L_1([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space L_Phi with Phi(x) = exp(x^2)-1.
In the first part of the paper we prove that for $2 < p, r < \infty$ every operator $T: L_p \to \ell_r$ is narrow. This completes the list of sequence and function Lebesgue spaces $X$ with the property that every operator $T:L_p \to X$ is narrow. Next, using similar methods we prove that every $\ell_2$-strictly singular operator from $L_p$, $1<p<\infty$, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator $T$ on $L_1[0,1]$ satisfies the assumption that for eac…
Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.
We introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice $E$ to a Banach space $Y$ when $E$ is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary $E$.
On the structure of variable exponent spaces
本文综述了变指数 Lebesgue 函数空间 $\lpv$ 的格结构，并研究了该空间之间严格奇异与不相交严格奇异算子，给出了包含映射不相交严格奇异性的新结果。
Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate two new approximation properties, which are strictly weaker than the bounded approximation property (BAP) for $\mathcal A$ of Lima, Lima and Oja (2010). We call them the weak BAP for $\mathcal A$ and the local BAP for $\mathcal A$, showing that the latter is in turn strictly weaker than the former. Under this framework, we address the question of approximation properties passing from dual spaces to underlying spaces. We relate the weak and local BAPs for $\mathcal A$ with approximation properties given by tensor norms and…
On an Analogue of a Property of Singular $M$-matrices, for the Lyapunov and the Stein Operators
本文证明了奇异不可约$M$-矩阵的一个性质（值域中唯一非负向量为零向量）在Lyapunov算子和Stein算子上的类比成立。
Quantitative Non-Compactness Properties of the Fourier Transform on Optimal Spaces
本文证明傅里叶变换在最优源空间与目标空间下不是严格奇异的，并给出傅里叶级数的类似结论，从而更精确刻画其非紧性程度。
We study a Bishop-Phelps-Bollobás type property for Banach space operators introduced by Dantas (2017). In that paper there is a local and a global version of a natural property which is somewhat similar but simpler compared to the Bishop-Phelps-Bollobás type property for operators studied in Acosta et al. (2008). Here we characterize the mentioned local property in the setting with strictly convex domain spaces and compact operators. We show that the local property implies that the domain space has strong convexity properties.
A property of strictly convex functions which differ from each other by a constant on the boundary of their domain
本文证明在自反实Banach空间中，对边界序列弱闭且非凸的闭凸集，任意稠密凸子集S中总存在一点，能作为所有在边界上与给定函数相差常数的严格凸下半连续函数的Gâteaux导数值集的代数内点。
The paper is concerned with the Bari basis property of a boundary value problem associated in $L^2([0,1]; \mathbb{C}^2)$ with the following $2 \times 2$ Dirac-type equation for $y = {\rm col}(y_1, y_2)$: $$L_U(Q) y =-i B^{-1} y' + Q(x) y = λy , \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad b_1 < 0 < b_2, $$ with a potential matrix $Q \in L^2([0,1]; \mathbb{C}^{2 \times 2})$ and subject to the strictly regular boundary conditions $Uy :=\{U_1, U_2\}y=0$. If $b_2 = -b_1 =1$ this equation is equivalent to one dimensional Dirac equation. We show that the system of root vectors $…
An arbitrary outward cuspidal domain is shown to be bi-Lipschitz equivalent to a Lipschitz outward cuspidal domain via a global transformation. This allows us to extend earlier Sobolev extension results on Lipschitz outward cuspidal domains from the work of Maz'ya and Poborchi to arbitrary outward cuspidal domains. We also establish a limit case of extension results on outward cuspidal domains.
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator $\mathcal{L}=-Δ+ \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the…
We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Ban…
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.
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