Unconditionally convergent multipliers and Bessel sequences

Can any unconditionally convergent multiplier be transformed to have the symbol (1) and Bessel sequences by shifting weights?

Multipliers are operators that combine (frame-like) analysis, a multiplication with a fixed sequence, called the symbol, and synthesis. They are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing. It is known that bounded symbols and Bessel sequences guarantee unconditional convergence. In this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers. In particular we show that, under mild conditions, unconditionally convergent multipliers can be transformed by shifting weights …

Unconditionally convergent series of operators and narrow operators on $L_1$

We introduce a class of operators on $L_1$ that is stable under taking sums of pointwise unconditionally convergent series, contains all compact operators and does not contain isomorphic embeddings. It follows that any operator from $L_1$ into a space with an unconditional basis belongs to this class.

加权框架、加权下半框架与无条件收敛乘子

Weighted frames, weighted lower semi frames and unconditionally convergent multipliers

本文研究序列在加权后成为框架或下半框架的条件，提出相应猜想并给出若干成立条件，同时证明两个关于无条件收敛乘子与权重平移的旧猜想之间的等价性，并揭示其与新猜想的关系。

When every polynomial is unconditionally converging

Letting $E$, $F$ be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator $E\rightarrow F$ is unconditionally converging, then every polynomial from $E$ to $F$ is unconditionally converging (definition as in the linear case). (2) If $E$ has the Dunford-Pettis property and every operator $E\rightarrow F$ is weakly compact, then every $k$-linear mapping from $E^k$ into $F$ takes weak Cauchy sequences into norm convergent sequences. In particular, every polynomial from $\ell_\infty$ into a space containing no copy of $\ell_\infty$ is completel…

通过强p-Cesàro可和性刻画赋范空间的完备性

Completeness of a Normed Space via Strong p-Cesàro Summability

本文利用强p-Cesàro可和性刻画赋范空间的完备性与桶状性，并给出弱无条件柯西级数与无条件收敛级数的新刻画。

Ideal boundedness of subseries and rearrangements in Banach spaces vs Banach spaces possessing a copy of $c_0$

Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are meager. We use Bessaga-Pełczy{ń}ski $c_0-$Theorem and concept of uniformly unconditionally bounded series. Moreover we prove similar result for the ideal boundedness for a class of Baire ideals using Talagrand's characterisation.