اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدعلى عوامل أندوس للدوال المنحنية والمنحنية العكسية
On Andos inequalities for convex and concave functions
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بالنسبة للمصفوفات الثنائية الموجبة النقطية $A$ و $B$، أثبت Ando و Zhan الإنحدارات $||| f(A)+f(B) ||| ge ||| f(A+B) |||$ و $||| g(A)+g(B) ||| le ||| g(A+B) |||$، لأي نوع من الأنظمة الثابتة المتحيزة، و لأي دالة تحفظ الأولوية السالبة على $[0,infty)$ بإخلاء دالة $g$. وقد تم توسيع هذه الإنحدارات حديثا إلى الدوال السالبة الغير موجبة والدوال المثبتة الغير منفية عن طريق Bourin و Uchiyama و Kosem على التوالي. في هذا البحث نناقش السؤال المرتبط بذلك حول ما إذا كانت الإنحدارات $||| f(A)-f(B) ||| le ||| f(|A-B|) |||$، و $||| g(A)-g(B) ||| ge ||| g(|A-B|) |||$، التي تم الحصول عليها من قبل Ando لدالة تحفظ الأولوية $f$ مع إخلاء $g$، يوجد لها توسيع مشابه إلى الدوال السالبة الغير موجبة والدوال المثبتة الغير منفية؟ ونجيب على هذا السؤال بشكل سلبي للمصفوفات العامة، وبشكل موجب للحالة الخاصة عندما $Age ||B||$. في طور هذا العمل، نقدم المفهوم الجديد للتحكم المغلوب بين الطيفين لمصفوفتين الهيرميتيتين، حيث يكون $Y$ نفسه مصفوفة هيرميتية، ونبرهن خاصية من هذا العلاق التي تسمح لتعزيز نتائج Bourin-Uchiyama و Kosem المذكورة أعلاه.
For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on $[0,infty)$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $||| f(A)-f(B) ||| le ||| f(|A-B|) |||$, and $||| g(A)-g(B) ||| ge ||| g(|A-B|) |||$, obtained by Ando, for operator monotone $f$ with inverse $g$, also have a similar generalisation to non-negative concave $f$ and convex $g$. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when $Age ||B||$. In the course of this work, we introduce the novel notion of $Y$-dominated majorisation between the spectra of two Hermitian matrices, where $Y$ is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.
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