arXiv subjects

Niushan Gao

Publications and source records attributed to Niushan Gao.

23 records · Page 2Linked to original sources

arXiv2015

Uo-convergence and its applications to Cesàro means in Banach lattices

A net $(x_α)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_α-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].

math.FA↗

arXiv2013

Unbounded Order Convergence and Application to Martingales without Probability

A net $(x_α)_{α\in Γ}$ in a vector lattice $X$ is unbounded order convergent (uo-convergent) to $x$ if $|x_α-x| \wedge y \xrightarrow{o} 0$ for each $y \in X_+$, and is unbounded order Cauchy (uo-Cauchy) if the net $(x_α-x_{α'})_{Γ\times Γ}$ is uo-convergent to 0. In the first part of this article, we study uo-convergent and uo-Cauchy nets in Banach lattices and use them to characterize Banach lattices with the positive Schur property and KB-spaces. In the second part, we use the concept of uo-Cauchy sequences to extend Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in $L_1(Ω;F)$ is almost surely uo-Cauchy in $F$, where $F$ is an order continuous Banach lattice with a weak unit.

math.FA↗

arXiv2012

Extensions of Perron-Frobenius Theory

The classical Perron-Frobenius theory asserts that for two matrices $A$ and $B$, if $0\leq B \leq A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$. This was recently extended in Bernik et al. (2012) to positive operators on $L_p(μ)$ with either $A$ or $B$ being irreducible and power compact. In this paper, we extend the results to irreducible operators on arbitrary Banach lattices.

math.FA↗

arXiv2012

On Commuting and Semi-commuting Positive Operators

Let $K$ be a positive compact operator on a Banach lattice. We prove that if either $[K>$ or $ =<K]=L_+(X)\cap {K}'$. We also establish the Perron-Frobenius Theorem for such operators $K$. Finally we apply the results to answer questions in Abramovich and Aliprantis (2002) and Bračič (2010).

math.FA↗

arXiv2012

Irreducible Semigroups of Positive Operators on Banach Lattices

The classical Perron-Frobenius theory asserts that an irreducible matrix $A$ has cyclic peripheral spectrum and its spectral radius $r(A)$ is an eigenvalue corresponding to a positive eigenvector. In Radjavi (1999) and Radjavi and Rosenthal (2000), this was extended to semigroups of matrices and of compact operators on $L_p$-spaces. We extend this approach to operators on an arbitrary Banach lattice $X$. We prove, in particular, that if $\iS$ is a commutative irreducible semigroup of positive operators on $X$ containing a compact operator $T$ then there exist positive disjoint vectors $x_1,...,x_r$ in $X$ such that every operator in $\iS$ acts as a positive scalar multiple of a permutation on $x_1,...,x_r$. Compactness of $T$ may be replaced with the assumption that $T$ is peripherally Riesz, i.e., the peripheral spectrum of $T$ is separated from the rest of the spectrum and the corresponding spectral subspace $X_1$ is finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator $T$, we show that $T$ is a cyclic permutation on $x_1,...,x_r$, $X_1=\Span{x_1,...,x_r}$, and if $S=\lim_j b_jT^{n_j}$ for some $(b_j)$ in $\mathbb R_+$ and $n_j\to\infty$ then $S=c(T_{|X_1})^k\oplus 0$ for some $c\ge 0$ and $0\le k<r$. We also extend results of Abramovich et al. (1992) and Grobler (1995) about peripheral spectra of irreducible operators.

math.FA↗