On operators whose adjoints or second adjoints attain their norms
Abstract
A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With NA1(X,Y) and NA2(X,Y) denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that NA2(c0,c0)=ℒ(c0,c0) and NA2(ℓ1,ℓ1)=ℒ(ℓ1,ℓ1), providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for c0, we undertake a systematic study of this equality within a natural family of ℓ1-preduals given by hyperplanes of c, obtaining a complete characterization in this setting. In particular, we prove that NA2(c,c)≠ ℒ(c,c), whereas NA3(c,c)=ℒ(c,c). We also establish Holub--Mujica-type theorems for the classes NA1 and NA2. More precisely, under suitable separability and approximation property assumptions, the identity ℒ(X,Y)=NA1(X,Y) forces every operator from X into Y to be compact, whereas ℒ(X,Y)=NA2(X,Y) forces every weakly compact operator from X into Y to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.
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BibTeX
@misc{dantas2026operators,
title = {{On operators whose adjoints or second adjoints attain their norms}},
author = {Sheldon Dantas and Mingu Jung and Miguel Martín},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00794},
eprint = {2609.00794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00794},
}
RIS
TY - UNPB
TI - On operators whose adjoints or second adjoints attain their norms
AU - Sheldon Dantas
AU - Mingu Jung
AU - Miguel Martín
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00794
UR - https://arxiv.org/abs/2609.00794
AB - A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With NA_1(X,Y) and NA_2(X,Y) denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that NA_2(c_0,c_0)=ℒ(c_0,c_0) and NA_2(ℓ_1,ℓ_1)=ℒ(ℓ_1,ℓ_1), providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for c_0, we undertake a systematic study of this equality within a natural family of ℓ_1-preduals given by hyperplanes of c, obtaining a complete characterization in this setting. In particular, we prove that NA_2(c,c)≠ ℒ(c,c), whereas NA_3(c,c)=ℒ(c,c). We also establish Holub--Mujica-type theorems for the classes NA_1 and NA_2. More precisely, under suitable separability and approximation property assumptions, the identity ℒ(X,Y)=NA_1(X,Y) forces every operator from X into Y to be compact, whereas ℒ(X,Y)=NA_2(X,Y) forces every weakly compact operator from X into Y to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.
ER -