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Differentiability of the operator norm on 
$\ell _p$ spaces
Published online by Cambridge University Press: 16 December 2024
Abstract
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 Fréchet 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 Fréchet 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.
Keywords
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
A portion of this research has been supported through the JC Bose National Fellowship and grant of Prof. Debashish Goswami, sponsored by the Science and Engineering Research Board (SERB) under the Government of India.
References
Abatzoglou, T. J.,
Norm derivatives on spaces of operators
. Math. Ann. 239(1979), no. 2, 129–135. MR51900810.1007/BF01420370CrossRefGoogle Scholar
Guerrero, J. B. and Palacios, A. R.,
Strong subdifferentiability of the norm on
${\mathrm{JB}}^{\ast }$
-triples
. Q. J. Math. 54(2003), no. 4, 381–390. MR203117110.1093/qmath/hag023CrossRefGoogle Scholar
${\mathrm{JB}}^{\ast }$
-triples
. Q. J. Math. 54(2003), no. 4, 381–390. MR203117110.1093/qmath/hag023CrossRefGoogle ScholarContreras, M. D.,
Strong subdifferentiability in spaces of vector-valued continuous functions
. Quart. J. Math. Oxford Ser. (2), 47(1996), no. 186, 147–155. MR139793410.1093/qmath/47.2.147CrossRefGoogle Scholar
Contreras, M. D., Payá, R. and Werner, W.,
${C}^{\ast }$
-algebras that are
$I$
-rings
. J. Math. Anal. Appl. 198(1996), no. 1, 227–236. MR137353710.1006/jmaa.1996.0078CrossRefGoogle Scholar
${C}^{\ast }$
-algebras that are
$I$
-rings
. J. Math. Anal. Appl. 198(1996), no. 1, 227–236. MR137353710.1006/jmaa.1996.0078CrossRefGoogle ScholarDantas, S., Jung, M., Roldán, Ó., and Zoca, A. R.,
Norm-attaining tensors and nuclear operators
. Mediterr. J. Math. 19(2022), no. 1, 38, 27. MR4371175CrossRefGoogle Scholar
Dantas, S., Kim, S. K., Lee, H. J., and Mazzitelli, M.,
Strong subdifferentiability and local Bishop-Phelps-Bollobás properties
. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(2020), no. 2, 47, 16. MR404987210.1007/s13398-019-00741-1CrossRefGoogle Scholar
Deeb, W. M. and Khalil, R. R.,
Exposed and smooth points of some classes of operation in
$L({l}^p)$
. J. Funct. Anal, 103(1992), no. 2, 217–228. MR115154710.1016/0022-1236(92)90120-8CrossRefGoogle Scholar
$L({l}^p)$
. J. Funct. Anal, 103(1992), no. 2, 217–228. MR115154710.1016/0022-1236(92)90120-8CrossRefGoogle ScholarDeville, R., Godefroy, G., and Zizler, V., Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR1211634Google Scholar
Diestel, J., Geometry of Banach spaces—selected topics, Springer-Verlag, Berlin, 1975. MR46109410.1007/BFb0082079CrossRefGoogle Scholar
Franchetti, C. and Payá, R.,
Banach spaces with strongly subdifferentiable norm
, Boll. Un. Mat. Ital. B (7), 7(1993), no. 1, 45–70. MR1216708Google Scholar
Godefroy, G., Indumathi, V., and Lust-Piquard, F.,
Strong subdifferentiability of convex functionals and proximinality
, J. Approx. Theory 116(2002), no. 2, 397–415. MR1911087CrossRefGoogle Scholar
Godefroy, G., Montesinos, V., and Zizler, V.,
Strong subdifferentiability of norms and geometry of Banach spaces
. Comment. Math. Univ. Carolin. 36(1995), no. 3, 493–502. MR1364490Google Scholar
Guerrero, J. B. and Peralta, A. M.,
Subdifferentiability of the norm and the Banach-Stone theorem for real and complex
${JB}^{\ast }$
-triples
. Manuscripta Math. 114(2004), no. 4, 503–516. MR2081949CrossRefGoogle Scholar
${JB}^{\ast }$
-triples
. Manuscripta Math. 114(2004), no. 4, 503–516. MR2081949CrossRefGoogle ScholarHarmand, P., Werner, D., and Werner, W.,
$M$
-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR1238713Google Scholar
$M$
-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR1238713Google ScholarKim, S. Kwang and Lee, H. J.,
Uniform convexity and Bishop-Phelps-Bollobás property
. Canad. J. Math. 66(2014), no. 2, 373–386. MR3176146Google Scholar
Kittaneh, F. and Younis, R.,
Smooth points of certain operator spaces
. Integral Equ. Oper. Theory 13(1990), no. 6, 849–855. MR1073855CrossRefGoogle Scholar
Lindenstrauss, J.,
On operators which attain their norm
. Israel J. Math. 1(1963), 139–148. MR160094CrossRefGoogle Scholar
Martínez-Moreno, J., Mena-Jurado, J. F., Payá-Albert, R., and Rodríguez-Palacios, Á.
An approach to numerical ranges without Banach algebra theory
. Illinois J. Math. 29(1985), no. 4, 609–626. MR80646910.1215/ijm/1256045498CrossRefGoogle Scholar
Oja, E. and Werner, D.,
Remarks on
$M$
-ideals of compact operators on
$X{\oplus}_pX$
. Math. Nachr. 152(1991), 101–111. MR112122710.1002/mana.19911520110CrossRefGoogle Scholar
$M$
-ideals of compact operators on
$X{\oplus}_pX$
. Math. Nachr. 152(1991), 101–111. MR112122710.1002/mana.19911520110CrossRefGoogle ScholarPellegrino, D. and Teixeira, E. V.,
Norm optimization problem for linear operators in classical Banach spaces
. Bull. Braz. Math. Soc. (N.S.) 40(2009), no. 3, 417–431. MR254051710.1007/s00574-009-0019-7CrossRefGoogle Scholar
Rao, T. S. S. R. K.,
Subdifferential set of an operator
. Monatsh. Math. 199(2022), no. 4, 891–898. MR449717310.1007/s00605-022-01739-5CrossRefGoogle Scholar
Rao, T. S. S. R. K.,
Subdifferentiability and polyhedrality of the norm
. Boll. Unione Mat. Ital. 16(2023), no. 4, 741–746. MR465343110.1007/s40574-023-00364-wCrossRefGoogle Scholar
Ruess, W. M. and Stegall, C. P.,
Extreme points in duals of operator spaces
. Math. Ann. 261(1982), no. 4, 535–546. MR68266510.1007/BF01457455CrossRefGoogle Scholar
Ruess, W. M. and Stegall, C. P.,
Exposed and denting points in duals of operator spaces
. Israel J. Math. 53(1986), no. 2, 163–190. MR845870CrossRefGoogle Scholar
Ryan, R. A., Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag, London, 2002. MR188830910.1007/978-1-4471-3903-4CrossRefGoogle Scholar
Taylor, K. F. and Werner, W.,
Differentiability of the norm in von Neumann algebras
. Proc. Amer. Math. Soc. 119(1993), no. 2, 475–480. MR114998010.1090/S0002-9939-1993-1149980-6CrossRefGoogle Scholar
Werner, W.,
Smooth points in some spaces of bounded operators
. Integral Equ. Oper. Theory 15(1992), no. 3, 496–502. MR115571710.1007/BF01200332CrossRefGoogle Scholar