Some Tensorial and Hadamard Product Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces
Abstract
Let $H$ be a Hilbert space. In this paper we show among others that, if $f$
is continuous differentiable convex on the open interval $I$ and $A,$ $B$
are selfadjoint operators in $B\left( H\right) $ with spectra $Sp( A) ,$ $Sp( B) \subset I,$ then we have the
tensorial inequality
\begin{align*}
\left( f^{\prime }\left( A\right) \otimes 1\right)\left( A\otimes1-1\otimes B\right)& \geq f\left(A\right) \otimes 1-1\otimes f\left(B\right) \\
& \geq \left( A\otimes 1-1\otimes B\right) \left( 1\otimes f^{\prime }\left(
B\right) \right)
\end{align*}
and the inequality for Hadamard product
\begin{align*}
\left( f^{\prime }\left( A\right) A\right) \circ 1-f^{\prime }\left(
A\right) \circ B& \geq \left[ f\left( A\right) -f\left( B\right) \right]
\circ 1 \\
& \geq A\circ f^{\prime }\left( B\right) -\left( f^{\prime }\left( B\right)
B\right) \circ 1.
\end{align*}.
References
- Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl., 26(1979), 203–241.
- Araki, H., Hansen, F., Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128(7)(2000), 2075–2084.
- Aujila, J.S., Vasudeva, H.L., Inequalities involving Hadamard product and operator means, Math. Japon., 42(1995), 265–272.
- Fujii, J.I., The Marcus-Khan theorem for Hilbert space operators, Math. Jpn., 41(1995), 531–535.
- Furuta, T., Micic Hot, J., Pecaric, J., Seo, Y., Mond-Pecaric Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
- Kitamura, K., Seo, Y., Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scient. Math.,1(2)(1998), 237–241.
- Koranyi, A., On some classes of analytic functions of several variables, Trans. Amer. Math. Soc., 101(1961), 520–554.
- Wada, S., On some refinement of the Cauchy-Schwarz Inequality, Lin. Alg. & Appl., 420(2007), 433–440.
Abstract
References
- Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl., 26(1979), 203–241.
- Araki, H., Hansen, F., Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128(7)(2000), 2075–2084.
- Aujila, J.S., Vasudeva, H.L., Inequalities involving Hadamard product and operator means, Math. Japon., 42(1995), 265–272.
- Fujii, J.I., The Marcus-Khan theorem for Hilbert space operators, Math. Jpn., 41(1995), 531–535.
- Furuta, T., Micic Hot, J., Pecaric, J., Seo, Y., Mond-Pecaric Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
- Kitamura, K., Seo, Y., Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scient. Math.,1(2)(1998), 237–241.
- Koranyi, A., On some classes of analytic functions of several variables, Trans. Amer. Math. Soc., 101(1961), 520–554.
- Wada, S., On some refinement of the Cauchy-Schwarz Inequality, Lin. Alg. & Appl., 420(2007), 433–440.
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Articles |
| Authors | |
| Publication Date | June 30, 2025 |
| Published in Issue | Year 2025 Volume: 17 Issue: 1 |