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Unconditionally convergent series of operators and narrow operators on \(L_1\). (English) Zbl 1077.46008

It has been shown by V. M. Kadets and R. V. Shvidkoy [Mat. Fiz. Anal. Geom. 6, No. 3–4, 253–263 (1999; Zbl 0960.46009)] that a pointwise unconditionally convergent series of compact operators \(T_n\colon L^1(0,1)\to X\) (\(X\) being a Banach space) is never an embedding operator. This is a strengthening of Pełczyński’s classical result that \(L^1\) cannot be embedded into a space with an unconditional basis.
In the present paper, the authors define \(\text{unc}\,({\mathcal U}) = \text{unc}_1({\mathcal U})\) (\({\mathcal U}\) a linear subspace of \({\mathcal L}(E,X)\), \(E\), \(X\) Banach spaces) as the set of all operators that can be represented as the sum of a pointwise unconditionally convergent series of operators from \({\mathcal U}\), and, for \(n\geq 1\): \[ \text{unc}_n({\mathcal U}) = \text{unc}\,(\text{unc}_{n-1} ({\mathcal U})). \] They show that if \(X=L^{p_1}\oplus_2 L^{p_2} \oplus_2 \cdots\), with \(1<p_n<\infty\) and \(p_n\to 1\), then \(\text{id}_X\in \text{unc}_2( {\mathcal K}(X)) \setminus \text{unc}_1( {\mathcal K}(X))\).
The main result is that there exists a linear subspace \({\mathcal U}_0\) of operators \(T\colon L^1\to X\) (called hereditarily PP-narrow operators) such that \({\mathcal K}(L^1,X)\subseteq {\mathcal U}_0\), \(\text{unc}\,({\mathcal U}_0)={\mathcal U}_0\), but no embedding operator is in \({\mathcal U}_0\). In particular, no embedding operator belongs to \(\text{unc}_n({\mathcal K}(L^1,X))\), for any \(n\). As a consequence, they recover an unpublished result of Rosenthal saying that \(L^1\) does not sign-embed into a space with an unconditional basis.
Several open questions conclude the paper.
Reviewer: Daniel Li (Lens)

MSC:

46B04 Isometric theory of Banach spaces
46B25 Classical Banach spaces in the general theory
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47B07 Linear operators defined by compactness properties

Citations:

Zbl 0960.46009
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