## 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:

  J. Bourgain and O. I. Reinov, On the approximation properties for the space $$H^{∞}$$, Math. Nachr. 122 (1985), 19–27. · Zbl 0584.46039  P. G. Casazza, C. L. García, and W. B. Johnson, An example of an asymptotically Hilbertian space which fails the approximation property, Proc. Amer. Math. Soc. 129 (2001), no. 10, 3017–3024. · Zbl 0983.46022  A. M. Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261–266. · Zbl 0267.46013  J. Dieudonné and L. Schwartz, La dualité dans les espaces $$(\mathcal F)$$ et $$\mathcal{LF}$$, Ann. Inst. Fourier Grenoble 1 (1949), 61–101.  T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197–200. · Zbl 0289.46015  A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16, 1955. · Zbl 0064.35501  M. \uI. Kadec and M. G. Snobar, Certain functionals on the Minkowski compactum (in Russian), Mat. Zametki 10 (1971), 453–458.  S. Kaijser and O. Reinov, Approximation properties of the space $$H^{∞}$$, Voronezh Winter Mathematical School “Modern methods of theory of functions and related problems” 2003, Jan 26–Feb 2. Voronezh: VGU (2003), 113–114.  J. Lindenstrauss, On James’ paper “Separable conjugate spaces”, Israel J. Math. 9 (1971), 279–284. · Zbl 0216.40802  J. Lindenstrauss and L.Tzafriri, Classical Banach spaces, vol.1: Sequence spaces, Berlin-Heidelberg-New York, 1977. · Zbl 0362.46013  F. Muñoz, E. Oja, and C. Piñeiro, On $$α$$-nuclear operators with applications to vector-valued function spaces, J. Funct. Anal. 269 (2015), 2871–2889. · Zbl 1332.46028  E. Oja, Grothendieck’s nuclear operator theorem revisited with an application to p-null sequences, J Funct. Anal. 263 (2012), 2876–2892. · Zbl 1301.47030  E. Oja and O. I. Reinov, Un contre-exemple à une affirmation de A.Grothendieck, C. R. Acad. Sc. Paris. – Serie I 305 (1987), 121–122. · Zbl 0621.46016  E. Oja and O. Reinov, A counterexample to A. Grothendieck, Proc. Acad. Est. SSR, Phys.-Math. 37 (1988), 14–17 (in Russian), Estonian and English summaries.  A. Pełczyński, Banach spaces of analytic functions and absolutely summing operators, AMS Regional Conference Series in Mathematics 30, Providence, 1977.  A. Pietsch, Operator ideals, North-Holland Mathematical Library, Vol. 20. Amsterdam - New York - Oxford: North-Holland Publishing Company, 1980.  G. Pisier, Estimations des distances à un espace euclidien et des constantes de projéction des espaces de Banach de dimensions finie, Seminaire d’Analyse Fonctionelle 1978-1979, Centre de Math., Ecole Polytech., Paris (1979), exposé 10, 1–21.  O. I. Reinov, Banach spaces without approximation property, Funct. Anal. Appl. 16 (1982), no. 4, 315–317.  O. I. Reinov, Approximation properties of order $$p$$ and the existence of non-$$p$$-nuclear operators with $$p$$-nuclear second adjoints, Math. Nachr. 109 (1982), 125–134. · Zbl 1231.47018  O. I. Reinov, A simple proof of two theorems of A. Grothendieck, Vestn. Leningr. Univ. 7 (1983), 115–116 (in Russian).  O. Reinov, A survey of some results in connection with Grothendieck approximation property, Math. Nachr. 119 (1984), 257–264. · Zbl 0532.46006  O. I. Reinov, Approximation properties $$AP_s$$ and $$p$$-nuclear operators (the case \(0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.