Geodesic
Approximation properties $\mathrm{AP}_s$ and $p$-nuclear operators (the case where $0$)
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 277-291

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Among other things, it is shown that there exist Banach spaces $Z$ and $W$ such that $Z^{**}$ and $W$ have bases, and for every $p\in[1,2)$ there is an operator $T\colon W\to Z$ that is not $p$-nuclear but $T^{**}$ is $p$-nuclear. 
@article{ZNSL_2000_270_a12,
     author = {O. I. Reinov},
     title = {Approximation properties $\mathrm{AP}_s$ and $p$-nuclear operators (the case where $0<s<1$)},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {277--291},
     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a12/}
}
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O. I. Reinov. Approximation properties $\mathrm{AP}_s$ and $p$-nuclear operators (the case where $0