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

Macphail's Theorem revisited

In 1947, M. S. Macphail constructed a series in $\ell_{1}$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach Space Theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky--Rogers Theorem asserts that in every infinite-dimensional Banach space $E$ there exists an unconditionally convergent series ${\textstyle\sum}x^{(j)}$ such that ${…

Ideal convergent subseries in Banach spaces

Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and $\sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(\mathcal{I}):=\left\{t \in \{0,1\}^{\mathbb{N}} \colon \sum_n t(n)x_n \textrm{ is } \mathcal{I}\textrm{-convergent}\right\}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure $λ$ on $\{0,1\}^{\mathb…

一般傅里叶级数的无条件收敛

Unconditional convergence of general Fourier series

本文研究一般正交规范系统下Lip_1函数傅里叶级数的无条件收敛条件，证明了最优条件的存在性，并发现任何ONS均含使无条件收敛成立的子系统。

Narrow operators on vector-valued sup-normed spaces

We characterise narrow and strong Daugavet operators on $C(K,E)$-spaces; these are in a way the largest sensible classes of operators for which the norm equation $\|Id+T\| = 1+\|T\|$ is valid. For certain separable range spaces $E$ including all finite-dimensional ones and locally uniformly convex ones we show that an unconditionally pointwise convergent sum of narrow operators on $C(K,E)$ is narrow, which implies for instance the known result that these spaces do not have unconditional FDDs. In a different vein, we construct two narrow operators on $C([0,1],\ell_1)$ whose sum is not narrow.

Frames containing a Riesz basis and preservation of this property under perturbation

Aldroubi has shown how one can construct any frame $\gtu$ starting with one frame $\ftu $,using a bounded operator $U$ on $l^2(N)$. We study the overcompleteness of the frames in terms of properties of $U$. We also discuss perturbation of frames in the sense that two frames are ``close'' if a certain operator is compact. In this way we obtain an equivalence relation with the property that members of the same equivalence class have the same overcompleteness. On the other hand we show that perturbation in the Paley-Wiener sense does not have this property. \\ Finally we construct a frame which i…