Almost-invariant and essentially-invariant halfspaces

Controlling almost-invariant halfspaces in both real and complex settings

If $T$ is a bounded linear operator acting on an infinite-dimensional Banach space $X$, we say that a closed subspace $Y$ of $X$ of both infinite dimension and codimension is an almost-invariant halfspace (AIHS) under $T$ whenever $TY\subseteq Y+E$ for some finite-dimensional subspace $E$, or, equivalently, $(T+F)Y\subseteq Y$ for some finite-rank perturbation $F:X\to X$. We discuss the existence of AIHS's for various restrictions on $E$ and $F$ when $X$ is a complex Banach space. We also extend some of these and other results in the literature to the setting where $X$ is a real Banach space i…

On the closedness of the sum of ranges of operators $A_k$ with almost compact products $A_i^* A_j$

Let $\mathcal{H}_1,\ldots,\mathcal{H}_n,\mathcal{H}$ be complex Hilbert spaces and $A_k:\mathcal{H}_k\to\mathcal{H}$ be a bounded linear operator with the closed range $Ran(A_k)$, $k=1,\ldots,n$. It is known that if $A_i^*A_j$ is compact for any $i\neq j$, then $\sum_{k=1}^n Ran(A_k)$ is closed. We show that if all products $A_i^*A_j$, $i\neq j$ are "almost" compact, then the subspaces $Ran(A_1),\ldots,Ran(A_n)$ are essentially linearly independent and their sum is closed.

Convolution invariant linear functionals and applications to summability methods

We study topologically invariant means on $L^{\infty}(\mathbb{R})$, the set of all essentially bounded functions on the real line, and prove that invariance with respect to a single convolution operator is sufficient for a mean to be topologically invariant. We also consider some applications of this result to summability methods. In particular, the notion of almost convergence is introduced for a function in $L^{\infty}(\mathbb{R})$, and a Tauberian theorem concerning almost convergence and a summability method defined by a Wiener kernel is obtained. Further, for the $C_{\infty}$ summability …

Optimality Conditions for Convex Stochastic Optimization Problems in Banach Spaces with Almost Sure State Constraints

We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty with an outlook for further applicatio…

关于序列贪心型基

On sequential greedy-type bases

本文通过固定序列$(a_n)_{n=1}^\infty$，比较阈值贪心算法与投影到该序列及其移位连续项的结果，提出$\mathcal{F}_{(a_n)}$-几乎贪心性质，揭示其与经典性质的等价条件，并建立相关类似结论及介于两类性质间的新条件。

Grand Lebesgue 空间中随机变量的可因子化收敛性

Factorizable convergence of random variables in Grand Lebesgue Spaces

本文研究随机变量几乎处处收敛的因子分解，将经典 Lebesgue-Riesz 空间中的结果推广至 Grand Lebesgue 空间，并给出参数的精确估计、实例及本质最优性证明。