توصيفگر ها :
نامساوي يانگ , نرم هيلبرت اشميت , نرم اثر , نرم ثابت يكاني
چكيده فارسي :
گسترش نامساويهاي عددي به عملگرها روي فضاي هيلبرت خصوصا در مورد ماتريسهاي n × n يكي از مباحث مهم
در نظريه عملگرها و آناليز ماتريسها ميباشد. پژوهشهاي زيادي در اين زمينه انجام شده است. همچنين شرط برقراري
تساوي در اين نامساويها مورد بررسي قرار گرفته است. از جمله نامساويهاي مهم، نامساوي يانگ و حالت خاص آن
ميانگين حسابي-هندسي است. براي اعداد حقيقي a و b و اعداد )∞ ,0( ∈ p, q كه 1 = 1 + 1 ، نامساوي يانگ بيان pq
|ab| ≤ 1|a|p + 1|b|q. pq
|a|1p|b|1q ≤1|a|+1|b|. pq
ميكند كه يا فرم معادل آن كه بهصورت زير است.
در حالت خاص اگر 2 = p = q و a,b اعداد حقيقي مثبت باشند، آنگاه ميانگين حسابي-هندسي حاصل ميشود كه بهصورت زير است
√ a+b
ab ≤
در اين پاياننامه نامساوي يانگ و معكوس آن به همراه نظريههاي مختلف آن را براي عملگرها روي فضاي هيلبرت
جداييپذير مورد بررسي قرار ميدهيم. ردهبندي موضوعي: 15A15 ،15A42 ،47A30 واژگان كليدي: نامساوي يانگ، نرم هيلبرت اشميت، نرم اثر، نرم ثابت يكاني.
چكيده انگليسي :
Abstract:
This M.Sc. thesis is based on the following papers
• Y, C. L, Y., Refinements and Reverses of Young Type Inequality, Journal of Math. Ineq. 14 (2020) 401–419.
• M, S. M., On Young and Heinz inequalities for faithful tracial states, Linear Multilinear Algebra, 65 (2017) 2432-2456.
The study of generalizing numerical inequalities to operators on Hilbert space, with a particular focus on matrices, constitutes a crucial area within operator theory and matrix analysis. Extensive research has been dedicated to ex- ploring this field, including investigations into the conditions for equality in these inequalities.
Among the significant inequalities in this context, Young’s inequality holds particular importance. Within this thesis, we undertake a comprehensive examination of Young’s inequality as it applies to matrices and operators on separable Hilbert space, delving into its various definitions and implications.
Through a meticulous analysis, we aim to deepen our understanding of Young’s inequality in the broader setting of operator theory. By extending its applicability to matrices and operators on Hilbert space, we contribute to the ongoing exploration of numerical inequalities and their implications in this fundamental domain.
One of the important inequalities in classical analysis is Young’s inequality. This inequality states that for two complex numbers z and w, and two numbers p and q in the interval (1, ∞) satisfying p1 + 1q = 1, we have:
|zw| ≤ 1|z|p + 1|w|q pq
Equality holds if and only if |z|p = |w|q.
In 1995, Ando proved that if A and B are positive semidefinite complex matrices of size n × n, then:
()
s(AB)≤s 1|A|p+1|B|q , iipq
where si(AB) denotes the i-th singular value of the matrix AB. The special case of p = q = 2 in the above inequality was examined by Bhatia and Kittaneh in 1990.
In 2002, Furuta et al. proved the above inequality for compact operators on a separable Hilbert space H. Herzallah and Kittaneh applied Young’s inequality to the Hilbert Schmidt norm and established the equality condition. They considered a weaker condition by replacing the equality in Young’s inequality with the equality of the Hilbert Schmidt norms in 2000.
Manjegani and Noroozi in 2015 proved the inverse of Young’s inequality for matrices. In recent years, various extensions of Young’s inequality for matrices and operators have been proven.
The first two chapters of this thesis provide the preliminaries for discussion matrix inequalities. In chapter 1, intro- duces numerical inequalities. Definitions and necessary properties related to matrices and operators are provided in Chapter 2. In chapter 3 presents theorems regarding the extension of Young’s inequality for matrices, and finally, Chapter 4 briefly discusses the generalization of these theorems for operators on Hilbert space.
