## 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 $$AP_{[c]}$$.

### 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.)

### Citations:

Zbl 0584.46039; Zbl 1323.47020
Full Text:

### References:

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