An Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Yıl 2024, Cilt: 6 Sayı: 1, 1 - 14, 03.05.2024

Sever Dragomır

https://doi.org/10.47087/mjm.1362713

Cited By: 1

Öz

Let H be a Hilbert space. Assume that f is continuously differentiable on I with ‖f′‖_{I,∞}:=sup_{t∈I}|f′(t)|<∞ and A, B are selfadjoint operators with Sp(A), Sp(B)⊂I, then

‖f((1-λ)A⊗1+λ1⊗B)-∫₀¹f((1-u)A⊗1+u1⊗B)du‖
≤‖f′‖_{I,∞}[(1/4)+(λ-(1/2))²]‖1⊗B-A⊗1‖

for λ∈[0,1]. In particular, we have the midpoint inequality

‖f(((A⊗1+1⊗B)/2))-∫₀¹f((1-u)A⊗1+u1⊗B)du‖
≤(1/4)‖f′‖_{I,∞}‖1⊗B-A⊗1‖.

Anahtar Kelimeler

Tensorial product, Selfadjoint operators, Operator norm, Ostrowski’s inequality

Kaynakça

• T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl. 26 (1979), 203-241.
• H. Araki and F. Hansen, Jensen's operator inequality for functions of several variables, Proc. Amer. Math. Soc. 128 (2000), No. 7, 2075-2084.
• J. S. Aujila and H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon. 42 (1995), 265-272.
• N. S. Barnett, P. Cerone and S. S. Dragomir, Some new inequalities for Hermite-Hadamard divergence in information theory. in Stochastic Analysis and Applications. Vol. 3, 7-19, Nova Sci. Publ., Hauppauge, NY, 2003. Preprint RGMIA Res. Rep. Coll. 5 (2002), No. 4, Art. 8, 11 pp. [Online https://rgmia.org/papers/v5n4/NIHHDIT.pdf]
• S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74(3)(2006), 417-478.
• S. S. Dragomir, Some tensorial Hermite-Hadamard type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 25 (2022), Art. 90, 14 pp. [Online https://rgmia.org/papers/v25/v25a90.pdf]
• A. Koranyi. On some classes of analytic functions of several variables. Trans. Amer. Math. Soc., 101 (1961), 520ñ554.
• A. Ebadian, I. Nikoufar and M. E. Gordji, Perspectives of matrix convex functions, Proc. Natl. Acad. Sci. USA, 108 (2011), no. 18, 7313-7314.
• J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators. Math. Jpn. 41 (1995), 531-535.
• T. Furuta, J. Micic Hot, J. Pecaric and Y. Seo, Mond-Pecaric Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
• K. Kitamura and Y. Seo, Operator inequalities on Hadamard product associated with Kadison's Schwarz inequalities, Scient. Math. 1 (1998), No. 2, 237-241.
• I. Nikoufar and M. Shamohammadi, The converse of the Loewner-Heinz inequality via perspective, Lin. & Multilin. Alg., 66 (2018), N0. 2, 243-249.
• A. Ostrowski, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.
• S. Wada, On some refinement of the Cauchy-Schwarz Inequality, Lin. Alg. & Appl. 420 (2007), 433-440.