Jian Shi, Zongsheng Gao
Let $A_{1}$, $A_{2}$, $...$, $A_{k}$ be strictly positive operators on a Hilbert space. This note is to show a sufficient condition of $A_{k}\geq A_{k-1}\geq\geq A_{3}\geq A_{2}\geq A_{1}$, which extends the related result before.
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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.
This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a premeasure limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a non-trivial way to ensure that the limit is a premeasure.
We study the Fourier characterisation of strictly positive definite functions on compact abelian groups. Our main result settles the case $G = F \times \mathbb{T}^r$, with $r \in \mathbb{N}$ and $F$ finite. The characterisation obtained for these groups does not extend to arbitrary compact abelian groups; it fails in particular for all torsion-free groups.
In this paper, we give a new approach to the theory of strictly positive kernels. Our method is based on the structure of Fock spaces. As its applications, various examples of strictly positive kernels are given. Moreover, we give a new proof of the universal approximation theorem for the Gauss kernel.
We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously obtained by D. Chen at all (2003) in the case in which the space is a sphere of dimension at least 2 and Menegatto at all (2006) in the case in which the space is the unit circle. As an application, we use the characterization to improve upon a recent result on the differentiability of positive definite kernels on the spaces.
Let $A_{i}\ (i=1, 2, ..., k)$ be bounded linear operators on a Hilbert space. This paper aims to show characterizations of operator order $A_{k}\geq A_{k-1}\geq...\geq A_{2}\geq A_{1}>0$ in terms of operator inequalities. Afterwards, an application of the characterizations is given to operator equalities due to Douglas's majorization and factorization theorem.
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.
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.
Necessary and sufficient conditions are given for the solvability of the operator valued two-variable autoregressive filter problem. In addition, in the two variable suboptimal Nehari problem sufficient conditions are given for when a strictly contractive little Hankel has a strictly contractive symbol.
Trees are very agreeable objects to work with, offering a diversity of behaviour within a structure that is sufficiently simple to admit precise analysis. Thus we are able to offer fairly satisfactory necessary and sufficient conditions on a tree $Υ$ for the existence of equivalent LUR or strictly convex norms on $\C_0(Υ)$ and for norms with the Kadec Property. In particular, we show that for a {\sl finitely branching} tree $Υ$ the space $\C_0(Υ)$ admits a Kadec renorming. Since some finitely branching trees fail the condition for strictly convex renormability, we obtain an example of a Banach…
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…
Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{n\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$ verifies the contraction property $¶(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$, where $S^\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random mat…
Invariant strictly convex renormings
本文研究Banach空间上关于群等距作用不变的严格凸等价范数的存在性，开发构造工具并证明保持性结果，同时肯定回答了关于$c$的全线性等距群不变严格凸重范化问题，并指出群顺从性的作用。
Strictly convex space : Strong orthogonality and Conjugate diameters
本文研究了严格凸空间中的强正交性与共轭直径，证明了严格凸的等价条件，并给出了Hamel基成为强正交基、范数下界估计及共轭直径存在性的充要条件。
Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.
We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of dimension greater than 1 and the other is not a circle. The result complements similar characterizations previously obtained for products of high dimensional spheres.
Let $\mathbb{H}\trianglelefteq\mathbb{G}$ be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with $L^{\infty}(\mathbb{G})$ that, roughly, measures the failure of $\mathbb{G}$ to act measure-preservingly on $\mathbb{H}$ by conjugation. The triviality of that element is equivalent to the condition that $\mathbb{G}$ and $\mathbb{G}/\mathbb{H}$ have the same modular element, by analogy with the classical situation. This condition is automatic if $\mathbb{H}\le \mathbb{G}$ is central, and in general implies the unimodularit…
A random copositive matrix is completely positive with positive probability
本文证明随维度增大，随机共正矩阵属于完全正矩阵锥的概率具有正下界，并推广到夹在两锥间的各类矩阵锥。
The bipolaron are two electrons coupled to the elastic deformations of an ionic crystal. We study this system in the Fröhlich approximation. If the Coulomb repulsion dominates, the lowest energy states are two well separated polarons. Otherwise the electrons form a bound pair. We prove the validity of the Pekar-Tomasevich energy functional in the strong coupling limit, yielding estimates on the coupling parameters for which the binding energy is strictly positive. Under the condition of a strictly positive binding energy we prove the existence of a ground state at fixed total momentum $P$, pro…
We determine the structure of all bijections on the cone of positive semidefinite operators which preserve the quantum $f$-divergence for an arbitrary strictly convex function $f$ defined on the positive halfline. It turns out that any such transformation is implemented by either a unitary or an antiunitary operator.
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