پژوهشهای ریاضی (Dec 2020)
On Subadditivity of Functions on Positive Operators Without Operator Monotonicity and Convexity
Abstract
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
