Some Properties of the Tsallis Relative Operator φ-Entropy
DOI:
https://doi.org/10.15407/mag17.02.216Анотація
У цій роботі ми вводимо поняття відносної операторної $\varphi$-ентропії Цалліса між двома суворо позитивними операторами і перевіряємо її властивості, такі як спільна опуклість, спільна субадитивність та монотонність. Ми також наводимо операторну нерівність типу Шеннона та обернену нерівність, які задовольняє відносна операторна $\varphi$-ентропія Цалліса.Mathematics Subject Classification: 81P45, 15A39, 47A63, 15A42, 81R15
Ключові слова:
перспективна функція, узагальнена перспективна функція, відносна операторна ентропія Цалліса, відносна операторна $\varphi$-ентропія ЦаллісаПосилання
M.-D. Choi, A Schwarz inequality for positive linear maps on C ∗ -algebras, Illinois J. Math. 18 (1974), 565–574. https://doi.org/10.1215/ijm/1256051007
A. Ebadian, I. Nikoufar, and M. Eshagi Gordji, Perspectives of matrix convex functions, Proc. Natl. Acad. Sci. 108 (2011), 7313–7314. https://doi.org/10.1073/pnas.1102518108
E.G. Effros, A matrix convexity approach to some celebrated quantum inequalities, Proc. Natl. Acad. Sci. 106 (2009), 1006–1008. https://doi.org/10.1073/pnas.0807965106
S.S. Dragomir, Lower bounds on partial subadditivity of noncommutative perspectives for operators in Hilbert spaces: the second variables, RGMIA Res. Rep. Coll. 23 (2020), Art. 127.
S. Furuichi, K. Yanagi, and K. Kuriyama, A note on operator inequalities of Tsallis relative operator entropy, Linear Algebra Appl. 407 (2005), 19–31. https://doi.org/10.1016/j.laa.2005.04.015
T. Furuta, J. Micic Hot, J.E. Pecaric and Y. Seo, Mond–Pecaric Method in Operator Inequalities, Element, Zagreb, 2005.
F. Hansen and G. Pedersen, Jensen’s Inequality for Operators and Löwner’s Theorem, Math. Ann. 258 (1982), 229–241. https://doi.org/10.1007/BF01450679
F. Hansen and G. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc. 35 (2003), 553–564. https://doi.org/10.1112/S0024609303002200
F. Hansen, The fast track to Lowner’s theorem, Linear Algebra Appl. 438 (2013), 4557–4571. https://doi.org/10.1016/j.laa.2013.01.022
F. Hiai and M. Mosonyi, Different quantum f -divergences and the reversibility of quantum operations, Rev. Math. Phys. 29 (2017), 1750023. https://doi.org/10.1142/S0129055X17500234
F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1979– 1980), 205–224. https://doi.org/10.1007/BF01371042
M.S. Moslehian, J. Mićić, and M. Kian, An operator inequality and its consequences, Linear Algebra Appl. 439 (2013), 584–591. https://doi.org/10.1016/j.laa.2012.08.005
I. Nikoufar, A. Ebadian, and M. Eshagi Gordji, The simplest proof of Lieb concavity theorem, Adv. Math. 248 (2013), 531–533. https://doi.org/10.1016/j.aim.2013.07.019
I. Nikoufar, On operator inequalities of some relative operator entropies, Adv. Math. 259 (2014), 376–383. https://doi.org/10.1016/j.aim.2014.03.019
I. Nikoufar, A perspective approach for characterization of Lieb concavity theorem, Demonstratio Math. 49 (2016), 463–469. https://doi.org/10.1515/dema-2016-0040
I. Nikoufar and M. Alinejad, Bounds of generalized relative operator entropies, Math. Inequal. Appl. 20 (2017), 1067–1078. https://doi.org/10.7153/mia-2017-20-68
I. Nikoufar and M. Shamohammadi, The converse of the Loewner–Heinz inequality via perspective, Linear Multilinear Algebra 66 (2018), 243–249. https://doi.org/10.1080/03081087.2017.1295432
I. Nikoufar, Convexity of parameter extensions of some relative operator entropies with a perspective approach, Glasgow Math. J. 62 (2020), 737–744. https://doi.org/10.1017/S0017089517000131
I. Nikoufar, A new characterization of the operator perspective, Linear Multilinear Algebra (2021), https://doi.org/10.1080/03081087.2021.1877247.
J. Pecaric, Power matrix means and related inequalities, Math. Commun. 1 (1996), 91–112.
O. E. Tikhonov, On matrix-subadditive functions and a relevant trace inequality, Linear Multilinear Algebra, 44 (1998), 25–28. https://doi.org/10.1080/03081089808818544
K. Yanagi, K. Kuriyama, and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Algebra Appl. 394 (2005), 109–118. https://doi.org/10.1016/j.laa.2004.06.025
K. Nishio and T. Ando, Characterizations of operations derived from network connections, J. Math. Anal. Appl. 53 (1976), 539–549. https://doi.org/10.1016/0022-247X(76)90090-1
M. Uchiyama, A. Uchiyama, and M. Giga, Superadditivity and derivative of operator functions, Linear Algebra Appl. 465 (2015), 401–411. https://doi.org/10.1016/j.laa.2014.09.021
L. Zou, Operator inequalities associated with Tsallis relative operator entropy, Math. Inequal. Appl. 18 (2015), 401–406. https://doi.org/10.7153/mia-18-29
