$\ well _{2}\oplus \mathcal {T}^{(2)}$的无条件基的唯一性

F. Albiac, J. L. Ansorena
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引用次数: 0

摘要

我们提供了准Banach格间紧算子的Pitt定理的一个新推广,它允许将Banach空间$\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$的有限直接和的无条件基描述为其和的无条件基的直接和。我们得到一般的分裂原理,特别是,如果每个$\mathbb{X}_{i}$有一个唯一的无条件基(直到等价和置换),那么$\mathbb{X}_{1}\oplus \cdots\oplus\mathbb{X}_{n}$也有一个唯一的无条件基。在我们的技术对巴拿赫和拟巴拿赫空间结构的新应用中,我们发现空间$\ell_2\oplus \mathcal{T}^{(2)}$具有唯一的无条件基。
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Uniqueness of unconditional basis of $\ell _{2}\oplus \mathcal {T}^{(2)}$
We provide a new extension of Pitt's theorem for compact operators between quasi-Banach lattices, which permits to describe unconditional bases of finite direct sums of Banach spaces $\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$ as direct sums of unconditional bases of its summands. The general splitting principle we obtain yields, in particular, that if each $\mathbb{X}_{i}$ has a unique unconditional basis (up to equivalence and permutation), then $\mathbb{X}_{1}\oplus \cdots\oplus\mathbb{X}_{n}$ has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $\ell_2\oplus \mathcal{T}^{(2)}$ has a unique unconditional basis.
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Corona Theorem. The Tomas–Stein inequality under the effect of symmetries Uniqueness of unconditional basis of $\ell _{2}\oplus \mathcal {T}^{(2)}$ Stability of solutions to some abstract evolution equations with delay Some more twisted Hilbert spaces
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