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**Some approximation properties and nuclear operators in spaces of analytical functions.**
*(English)*
Zbl 1404.46017

Recall that a Banach space \(X\) has the approximation property (AP) if the identity operator in \(X\) can be approximated by finite rank operators in the topology of uniform convergence on compact sets. Since every compact set is contained in the closed absolutely convex hull of a sequence which converges to zero, the AP can be reformulated as follows: A Banach space \(X\) has the AP if, for every sequence \((x_n)_n\subset X\) with \(\|x_n\| \longrightarrow0\) as \(n \to \infty\) and \(\varepsilon >0\), there exists a finite rank operator \(R\) in \(X\) such that \(\sup_{n \in \mathbb N}\|x_n-Rx_n\|<\varepsilon\).

The authors take this last form of the AP and introduce a weak version of the AP: Denote by \(\mathcal C\) the set of all real positive sequences tending to \(\infty\). Then, for a sequence \(c=(c_n)_n \in \mathcal C\), a Banach space \(X\) has the approximation property with respect to \(c\) (\(AP_{[c]}\)) if, for every \(\varepsilon >0\) and any sequence \((x_n)_n\) in \(X\) with \(\|x_n\|\leq c_n^{-1}\), there exists a finite rank operator \(R\) in \(X\) such that \(\|Rx_n-x_n\| \leq \varepsilon\) for every \(n\). Also, if \(\mathcal C_0\) is a subset of \(\mathcal C\), \(X\) has the approximation property respect to \(\mathcal C_0\) \((AP_{[\mathcal C_0]})\) if it has the \(AP_{[c]}\) for all \(c \in \mathcal C_0\).

These type of approximation properties generalize some AP’s considered in different works. For instance, in [J. Bourgain and O. Reinov, Math. Nachr. 122, 19–27 (1985; Zbl 0584.46039)], it was shown that \(H^{\infty}\) (the space of bounded analytic functions in the unit disk) has the \(AP_{[\log(n)]}\) and, if \(\ell_p^{-1}=\{(c_n)_n \in \mathcal C : (c_n^{-1})_n \in \ell_p\}\), then the \(AP_{\ell_p^{-1}}\) coincide with the \(AP_s\) considered, among others, in [O. I. Reinov, J. Math. Anal. Appl. 415, No. 2, 816–824 (2014; Zbl 1323.47020)], where \(\frac 1s=2-\frac 1p\). Also, note that the \(AP_{[\mathcal C]}\) is exactly the \(AP\).

Several characterizations of the \(AP_{[c]}\) are given and some results are applied to \(L_p\)-spaces, to \(H^{\infty}\) and its predual. Also, the authors give a nice approach in the study of whether the nuclear operators are a regular operator ideal or not, which is one of the main results of the article. Let me explain this a little:

Recall that an operator \(S: X\rightarrow Y\) is nuclear if there exist sequences \((x'_n)_n \subset X'\) and \((y_n)_n \subset Y\) with \(\sum_{n=1}^{\infty} \|x_n'\|\|y_n\| < \infty\) such that \(S=\sum_{n=1}^{\infty} x'_n\otimes y_n\). Let \(J_Y: Y\rightarrow Y''\) be the natural injection of \(Y\) into its second dual and take an operator \(T: X\rightarrow Y\). If \(J_Y\circ T: X\rightarrow Y''\) is nuclear, then must \(T\) be nuclear? When \(X'\) or \(Y'''\) has the \(AP\), the answer is yes and the hypothesis over \(X\) or \(Y\) is sharp.

Here, the authors introduce, for a sequence \(c \in \mathcal C\), the \([c]\)-nuclear operators from \(X\) to \(Y\) as those \(T\) which admit a representation of the form \(Tx=\sum_{n=1}^{\infty} \mu_n x'_n(x) y_n\) for \(x \in X\), where \((x'_n)_n \subset X'\) and \((y_n)_n \subset Y\) with \(\sum_{n=1}^{\infty} \|x_n\|\|y_n\| < \infty\) and \(|\mu_n|\leq 1/c_n\). Then, they show that an operator \(T: X\rightarrow Y\) such that \(J_Y\circ T: X\rightarrow Y''\) is \([c]\)-nuclear is nuclear itself if \(X'\) or \(Y'''\) has the \(AP_{[c]}\).

The authors take this last form of the AP and introduce a weak version of the AP: Denote by \(\mathcal C\) the set of all real positive sequences tending to \(\infty\). Then, for a sequence \(c=(c_n)_n \in \mathcal C\), a Banach space \(X\) has the approximation property with respect to \(c\) (\(AP_{[c]}\)) if, for every \(\varepsilon >0\) and any sequence \((x_n)_n\) in \(X\) with \(\|x_n\|\leq c_n^{-1}\), there exists a finite rank operator \(R\) in \(X\) such that \(\|Rx_n-x_n\| \leq \varepsilon\) for every \(n\). Also, if \(\mathcal C_0\) is a subset of \(\mathcal C\), \(X\) has the approximation property respect to \(\mathcal C_0\) \((AP_{[\mathcal C_0]})\) if it has the \(AP_{[c]}\) for all \(c \in \mathcal C_0\).

These type of approximation properties generalize some AP’s considered in different works. For instance, in [J. Bourgain and O. Reinov, Math. Nachr. 122, 19–27 (1985; Zbl 0584.46039)], it was shown that \(H^{\infty}\) (the space of bounded analytic functions in the unit disk) has the \(AP_{[\log(n)]}\) and, if \(\ell_p^{-1}=\{(c_n)_n \in \mathcal C : (c_n^{-1})_n \in \ell_p\}\), then the \(AP_{\ell_p^{-1}}\) coincide with the \(AP_s\) considered, among others, in [O. I. Reinov, J. Math. Anal. Appl. 415, No. 2, 816–824 (2014; Zbl 1323.47020)], where \(\frac 1s=2-\frac 1p\). Also, note that the \(AP_{[\mathcal C]}\) is exactly the \(AP\).

Several characterizations of the \(AP_{[c]}\) are given and some results are applied to \(L_p\)-spaces, to \(H^{\infty}\) and its predual. Also, the authors give a nice approach in the study of whether the nuclear operators are a regular operator ideal or not, which is one of the main results of the article. Let me explain this a little:

Recall that an operator \(S: X\rightarrow Y\) is nuclear if there exist sequences \((x'_n)_n \subset X'\) and \((y_n)_n \subset Y\) with \(\sum_{n=1}^{\infty} \|x_n'\|\|y_n\| < \infty\) such that \(S=\sum_{n=1}^{\infty} x'_n\otimes y_n\). Let \(J_Y: Y\rightarrow Y''\) be the natural injection of \(Y\) into its second dual and take an operator \(T: X\rightarrow Y\). If \(J_Y\circ T: X\rightarrow Y''\) is nuclear, then must \(T\) be nuclear? When \(X'\) or \(Y'''\) has the \(AP\), the answer is yes and the hypothesis over \(X\) or \(Y\) is sharp.

Here, the authors introduce, for a sequence \(c \in \mathcal C\), the \([c]\)-nuclear operators from \(X\) to \(Y\) as those \(T\) which admit a representation of the form \(Tx=\sum_{n=1}^{\infty} \mu_n x'_n(x) y_n\) for \(x \in X\), where \((x'_n)_n \subset X'\) and \((y_n)_n \subset Y\) with \(\sum_{n=1}^{\infty} \|x_n\|\|y_n\| < \infty\) and \(|\mu_n|\leq 1/c_n\). Then, they show that an operator \(T: X\rightarrow Y\) such that \(J_Y\circ T: X\rightarrow Y''\) is \([c]\)-nuclear is nuclear itself if \(X'\) or \(Y'''\) has the \(AP_{[c]}\).

Reviewer: Pablo Turco (Buenos Aires)

### MSC:

46B28 | Spaces of operators; tensor products; approximation properties |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |

### Keywords:

nuclear operator; tensor product; approximation property; space of bounded analytical functions
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\textit{S. Kaijser} and \textit{O. I. Reinov}, Adv. Oper. Theory 4, No. 1, 265--283 (2019; Zbl 1404.46017)

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