PreprintCC BY-NC-ND 4.0arXiv
Hereditary Approximation under Hilbertian Extension
Abstract
In this paper, we prove that if a Banach space X has the hereditary approximation property, then X⊕2 H has the hereditary approximation property for every Hilbert space H. This answers a question of Johnson and Szankowski. In addition, if X is also reflexive, then every closed subspace of X⊕2H has the metric approximation property. The proof uses Grothendieck's characterization of the approximation property, a factorization of nuclear operators through quotients of X, and a nuclear perturbation of the coordinate projection on a closed graph.
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BibTeX
@misc{cai2026hereditary,
title = {{Hereditary Approximation under Hilbertian Extension}},
author = {Gang Cai and Qingjin Cheng and Longfa Sun and Yinghua Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41758},
eprint = {2610.41758},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41758},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Hereditary Approximation under Hilbertian Extension
AU - Gang Cai
AU - Qingjin Cheng
AU - Longfa Sun
AU - Yinghua Sun
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41758
UR - https://arxiv.org/abs/2610.41758
AB - In this paper, we prove that if a Banach space X has the hereditary approximation property, then X⊕_2 H has the hereditary approximation property for every Hilbert space H. This answers a question of Johnson and Szankowski. In addition, if X is also reflexive, then every closed subspace of X⊕_2H has the metric approximation property. The proof uses Grothendieck's characterization of the approximation property, a factorization of nuclear operators through quotients of X, and a nuclear perturbation of the coordinate projection on a closed graph.
ER -