Relative functional entropy in convex analysis

• Autorzy: Raissouli M.
• Języki publikacji: EN

Abstrakty

EN

The purpose of this work is to extend the relative entropy S(A|B) = A 1/2 log(A -1/2 BA -1/2)A 1/2 from positive operators to convex functionals. Our functional approach implies immediately, in a fast way, some simplifications and improvements for that of positive operators already discussed in the literature.

Słowa kluczowe

EN

convex analysis; relative operator entropy; power harmonic operator and functional means; relative functional entropy

PL

analiza wypukła; entropia względna; entropia relatywna

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2011
• Tom: Vol. 17, nr 2
• Strony: 231--239
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Raissouli M.

• Department of Mathematics, Faculty of Sciences, Taibah University, Al Madinah Al Munawwarah, Kingdom of Saudi Arabia,

Bibliografia

• [1] M. Atteia and M. Raissouli, Self dual operator on convex functionals, geometric mean and square root of convex functionals, J. Convex Anal. 8 (2001), 223-240.
• [2] J. P. Aubin, Analyse non lineaire et ses motivations economiques, Masson, Paris, 1983.
• [3] H. Brezis. Analyse fonctionnelle: Theorie et applications, Masson, Paris. 1983.
• [4] A. El Biari, R. Ellaia and M. Raissouli, Stability of geometric and harmonic functional means, J. Convex Anal. 10 (2003), no. 1, 199-210.
• [5] J. I. Fujii, Kubo-Ando theory of convex functional means, Sci. Math. Jpn. 7 (2002), 299-311.
• [6] J. I. Fujii, M. Fujii and Y. Seo, An extension of the Kubo-Ando theory. Solidarities, Math. Jap. 35 (1990), 387-396.
• [7] J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Jap. 34 (1989), 341-348.
• [8] T. Futura, Simple proof of the concavity of operator entropy f(A) = -A log A, Math. Inequal. Appl. 3 (2000), no. 2, 305-306.
• [9] T. Futura, M. Giga and M. Yanagida, Simple proof of jointly concavity of the relative operator entropy S(A|B) = A1/2 log (A-1/2 BA-1/2) A1/2, Math. Inequal. Appl. 6 (2003), no. 4, 713-714.
• [10] M. Nakamura and H. Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad. 37(1961), 149-154.
• [11] M. Raissouli, Logarithmic functional mean in convex analysis, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Art. 102.
• [12] M. Raissouli and H. Bouziane, Functional logarithm in the sense of convex analysis, J. Convex Anal. 10 (2003), no. 1, 229-244.
• [13] M. Raissouli and H. Bouziane, Functional limited development in convex analysis, Ann. Sci. Math. Quebec 29 (2005), no. 1, 97-110.
• [14] M. Raissouli and H. Bouziane, Arithmetico-geometrico-harmonic functional mean in the sense of convex analysis, Ann. Sci. Math. Quebec 3 (2006), no. 1, 79-107.
• [15] M. Raissouli and M. Chergui, Arithmetico-geometric and geometrico-harmonic means of two convex functionals, Sci. Math. Jpn. 55 (2002), no. 3,485-492.