Volume 6, Issue 4 (Vol. 6, No. 4, 2020)                   mmr 2020, 6(4): 521-526 | Back to browse issues page

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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