On Subadditivity of Functions on Positive Operators Without Operator Monotonicity and Convexity
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 tha...
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| Format: | Article |
| Language: | fas |
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Kharazmi University
2020-12-01
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| Series: | پژوهشهای ریاضی |
| Subjects: | |
| Online Access: | http://mmr.khu.ac.ir/article-1-2931-en.html |
| Summary: | 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 |
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| ISSN: | 2588-2546 2588-2554 |