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.
The structure of non-compactness of optimal Sobolev embeddings of $m$-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular.
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.
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.
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.
In this paper, we study the existence of infinite dimensional closed linear subspaces of a rearrangement invariant space on [0,1] every nonzero element of which does not belong to any included rearrangement invariant space of the same class such that the inclusion operator is disjointly strictly singular. We consider Lorentz, Marcinkiewicz and Orlicz spaces. The answer is affirmative for Marcinkiewicz spaces and negative for Lorentz and Orlicz spaces. Also, the same problem is studied for Nakano spaces assuming different hypothesis.
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…
Classes of strictly singular operators and their products
V. D。Milman在\cite{Milman:70}中证明了两个严格奇异算子在$L_p[0,1]$（$1\le p<\infty$）或$C[0,1]$上的乘积是紧的。本文利用Schreier族$§_ξ$定义了$§_ξ$ -严格奇异算子类，并改进了Milman的技巧，证明了该类算子的某些乘积是紧的，假设基础巴拿赫空间有1000个Schreier扩展序列的等价类.最后定义了一类$§_ξ$ -遗传不可分解的巴拿赫空间，并研究了其上的算子.
Multiplication operators on L(L_p) and $\ell_p$-strictly singular operators
给出了L（L_p）上弱紧乘法算子的一个分类：$1<p<\infty$.这回答了Saksman和Tylli在1992年提出的一个问题。主要结果是：如果L_p（1<p<2）上的算子T是$\ell_p$-严格奇异的，且T_{|X}是L_p的某个子空间X的同构，则X对所有r<2嵌入L_r，但X不必同构于希尔伯特空间。证明了若T是Cantor群$1\le p <2$的L_p上有偏硬币的卷积，且$T_{|X}$是L_p的某个自反子空间X的同构，则X同构于一个希尔伯特空间. p=1的情况回答了罗森塔尔在1976年提出的一个问题。
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 study ``disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of Köthe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive Köthe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply sup…
We prove weighted estimates for singular integral operators which operate on function spaces on a half-line. The class of admissible weights includes Muckenhoupt weights and weights satisfying Sawyer's one-sided conditions. The kernels of the operators satisfy relaxed Dini conditions. We apply the weighted estimates to extrapolation of maximal $L^p$ regularity of first order, second order and fractional order Cauchy problems into weighted rearrangement invariant Banach function spaces. In particular, we provide extensions, as well as a unification of recent results due to Auscher and Axelsson,…
We study operators on the Kalton-Peck Banach space $Z_2$ from various points of view: matrix representations, examples, spectral properties and operator ideals. For example, we prove that there are non-compact, strictly singular operators acting on $Z_2$, but the product of two of them is a compact operator. Among applications, we show that every copy of $Z_2$ in $Z_2$ is complemented, and each semi-Fredholm operator on $Z_2$ has complemented kernel and range, the space $Z_2$ is $Z_2$-automorphic and we give a partial solution to a problem of Johnson, Lindenstrauss and Schetchman about strictl…
An operator $T$ from a Banach lattice $E$ into a Banach space is disjointly non-singular ($DN$-$S$, for short) if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for $DN$-$S$ operators, including a perturbative characterization. For $E=L_p$ ($1< p<\infty$) we improve the results, and we show that the $DN$-$S$ operators have a different behavior in the cases $p=2$ and $p\neq 2$. As an application we prove that the strongly embedded subspaces of $L_p$ form an open subset in the set of all closed subspaces.
We show that the perturbation class for the upper semi-Fredholm operators between two Banach spaces X and Y coincides with the strictly singular operators when X is subprojective and that the perturbation class for the lower semi-Fredholm operators coincides with the strictly cosingular operators when Y is superprojective. Similar results were previously obtained under stronger conditions for X and Y.
In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{ω_φ}$, where $ω_φ= e^{-φ}$ and $φ$ is a subharmonic function. We consider compact Hankel operators $H_{\overline ϕ}$, with anti-analytic symbols ${\overline ϕ}$, and give estimates of the trace of $h(|H_{\overline ϕ}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{\overline ϕ}))_n$ in terms of decreasing rearrangement of $|ϕ'|/\sqrt{Δφ}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{\overline ϕ…
