Abstract
Let $H$ be a Hilbert space. In this paper we show among others that, if $f,$ $g$ are synchronous and continuous on $I$ and $A,$ $B$ are selfadjoint with spectra ${Sp}\left( A\right) ,$ ${Sp}\left( B\right) \subset I,$ then%
\begin{equation*}
\left( f\left( A\right) g\left( A\right) \right) \otimes 1+1\otimes \left(
f\left( B\right) g\left( B\right) \right) \geq f\left( A\right) \otimes
g\left( B\right) +g\left( A\right) \otimes f\left( B\right)
\end{equation*}%
and the inequality for Hadamard product%
\begin{equation*}
\left( f\left( A\right) g\left( A\right) +f\left( B\right) g\left( B\right)
\right) \circ 1\geq f\left( A\right) \circ g\left( B\right) +f\left(
B\right) \circ g\left( A\right) .
\end{equation*}%
Let either $p,q\in \left( 0,\infty \right) $ or $p,q\in \left( -\infty
,0\right) $. If $A,$ $B>0,$ then
\begin{equation*}
A^{p+q}\otimes 1+1\otimes B^{p+q}\geq A^{p}\otimes B^{q}+A^{q}\otimes B^{p},
\end{equation*}%
and%
\begin{equation*}
\left( A^{p+q}+B^{p+q}\right) \circ 1\geq A^{p}\circ B^{q}+A^{q}\circ B^{p}.
\end{equation*}
References
- [1] H. Araki, F. Hansen, Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128 (7) (2000), 2075-2084.
- [2] A. Koranyi, On some classes of analytic functions of several variables, Trans. Amer. Math. Soc., 101 (1961), 520-554.
- [3] T. Furuta, J. Micic Hot, J. Pecaric, Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
- [4] S. Wada, On some refinement of the Cauchy-Schwarz inequality, Lin. Alg. & Appl., 420 (2007), 433-440.
- [5] J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators, Math. Jpn., 41 (1995), 531-535.
- [6] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl., 26 (1979), 203-241.
- [7] J. S. Aujila, H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon., 42 (1995), 265-272.
- [8] K. Kitamura, Y. Seo, Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scient. Math. 1(2) (1998), 237-241.
- [9] P. Bhunia, K. Paul, A. Sen, Numerical radius inequalities for tensor product of operators, Proc. Indian Acad. Sci. (Math. Sci.), 133(3) (2023).
- [10] H. L. Gau, K. Z. Wang, P. Y. Wu, Numerical radii for tensor products of operators, Integr. Equ. Oper. Theory, 78 (2014), 375–382.
Abstract
References
- [1] H. Araki, F. Hansen, Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128 (7) (2000), 2075-2084.
- [2] A. Koranyi, On some classes of analytic functions of several variables, Trans. Amer. Math. Soc., 101 (1961), 520-554.
- [3] T. Furuta, J. Micic Hot, J. Pecaric, Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
- [4] S. Wada, On some refinement of the Cauchy-Schwarz inequality, Lin. Alg. & Appl., 420 (2007), 433-440.
- [5] J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators, Math. Jpn., 41 (1995), 531-535.
- [6] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl., 26 (1979), 203-241.
- [7] J. S. Aujila, H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon., 42 (1995), 265-272.
- [8] K. Kitamura, Y. Seo, Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scient. Math. 1(2) (1998), 237-241.
- [9] P. Bhunia, K. Paul, A. Sen, Numerical radius inequalities for tensor product of operators, Proc. Indian Acad. Sci. (Math. Sci.), 133(3) (2023).
- [10] H. L. Gau, K. Z. Wang, P. Y. Wu, Numerical radii for tensor products of operators, Integr. Equ. Oper. Theory, 78 (2014), 375–382.
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | November 7, 2023 |
| Publication Date | December 25, 2023 |
| Submission Date | September 19, 2023 |
| Acceptance Date | October 31, 2023 |
| Published in Issue | Year 2023 Volume: 6 Issue: 4 |
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