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anjidani E. On Subadditivity of Functions on Positive Operators Without Operator Monotonicity and Convexity. mmr 2020; 6 (4) :521-526
URL: http://mmr.khu.ac.ir/article-1-2931-en.html
University of Neyshabur , ehsan.mathematics@gmail.com
Abstract:   (1200 Views)

‎‎‎‎In ‎this ‎paper, ‎we ‎investigate ‎the ‎subadditivity ‎of ‎functions ‎on positive ‎operators ‎without ‎operator ‎monotonicity ‎and ‎operator ‎convexity: Let ‎‎$‎A‎$ ‎and ‎$‎B‎$ ‎be positive operators on a Hilbert space ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎$‎‎mathcal{H}‎$ ‎satisfying‎ ‎‎$‎0leq AB+BA‎$. Suppose that for the operator

‎‎‎‎$$‎E=(A+B)^{-frac{1}{2}}left(A^2+B^2right)(A+B)^‎{‎-frac{1}{2}}‎,$$‎‎

the open interval ‎$‎(m_E,M_E)‎$ where, ‎$‎m‎_E$ ‎and ‎‎$‎M_E‎$ ‎are ‎bounds ‎of ‎operator ‎‎$‎E‎$‎,‎ ‎does ‎not ‎intersect ‎the ‎spectrums ‎of ‎operators ‎‎$‎A‎$ ‎‎and ‎‎$‎B‎$‎.‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎

Then, for every continuous function ‎‎‎‎$‎g:(0,infty)‎rightarrow‎‎mathbb{R}^+‎$ ‎for ‎which ‎the function‎ ‎‎$‎f(t)=frac{g(t)}{t}‎$ is convex and decreasing, we have ‎‎‎

‎‎$‎‎$‎g(A+B)leq c(m,M,f)(g(A)+g(B)),‎$‎‎$‎‎‎

where, ‎$‎m‎$ ‎and ‎‎$‎M‎$ ‎are ‎bounds ‎of ‎operator ‎‎$‎A+B‎$ ‎and‎‎‎‎‎

‎‎$‎‎$‎‎c(m,M,f):=max_{mleq tleq M}left{frac{‎frac{f(M)-f(m)}{M-m}t+‎frac{Mf(m)-mf(M)}{M-m}}{f(t)‎}right}‎.‎$‎‎$‎./files/site1/files/64/3.pdf

Full-Text [PDF 517 kb]   (312 Downloads)    
Type of Study: S | Subject: alg
Received: 2019/03/22 | Revised: 2021/02/13 | Accepted: 2019/07/17 | Published: 2021/01/29 | ePublished: 2021/01/29

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