AI 中文总结
研究\(\ell_p\)空间上有界线性算子范数的可微性，给出强次可微性刻画、特定集合稠密性证明及弗雷歇可微性刻画，证明了该空间上算子范数的弗雷歇可微性与加托可微性一致，推广了希尔伯特空间相关定理。
AI 中文摘要
本文给出了\(1\leq p<\infty\)时，\(\ell_p\)空间上有界线性算子范数的强次可微性的一个刻画。此外，我们证明了\({B}(\ell_p, \ell_q)\)中范数是强次可微的所有有界线性算子的集合在\({B}(\ell_p, \ell_q)\)中是稠密的。另外，我们给出了\(1 < p, q < \infty\)时，从\(\ell_p\)到\(\ell_q\)的有界线性算子范数的弗雷歇可微性的一个刻画。应用此结果，我们将表明\(\ell_p\)空间上有界线性算子范数的弗雷歇可微性和加托可微性是一致的，推广了关于希尔伯特空间上算子范数的一个已知定理。
英文摘要
In this paper, we present a characterization of strong subdifferentiability of the norm of bounded linear operators on $\ell_p$ spaces, $1\leq p<\infty$. Furthermore, we prove that the set of all bounded linear operators in ${B}(\ell_p, \ell_q)$ for which the norm of ${B}(\ell_p, \ell_q)$ is strongly subdifferentiable is dense in ${B}(\ell_p, \ell_q)$. Additionally, we present a characterization of Frechet differentiability of the norm of bounded linear operators from $\ell_p$ to $\ell_q$, where $1 < p, q < \infty$. Applying this result, we will show that the Frechet differentiability and the Gateaux differentiability of the norm of bounded linear operators on $\ell_p$ spaces coincide, extending a known theorem regarding the operator norm on Hilbert spaces.
CommentsPublished in Canadian journal of Mathematics
Journal refCanadian Journal of Mathematics. Published online 2024:1-20
