Inequalities involving Hadamard product for sector matrices

Document Type : Original Article

Authors
Hamideh Mohammadzadehkan 1
Somayeh Malekinejad 2
1 Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
2 Department of Mathematics, Payame Noor University, P.O. Box 19395-3697
10.22034/maco.5.1.4
Abstract
In this paper, we investigate new inequalities for the Hadamard product of sector matrices. Using operator theory techniques and concepts of operator means, we establish new inequalities for the Hadamard product in this class of matrices. In particular, for matrices \( A_i, B_i \in S_{\theta} \) satisfying 
    \(
    I(A_i \sigma B_i) \circ I(A_i \sigma^{\perp} B_i) \leq 0,
    \)
    the following inequality holds:
    \begin{align*}
        \left( \sum_{i=1}^{k} R(A_i \# B_i) \right) &\circ \left( \sum_{i=1}^{k} R(A_i \# B_i) \right)\\
        & \leq \sec^4(\theta) R \left( \left( \sum_{i=1}^{k} A_i \sigma B_i \right) \circ \left( \sum_{i=1}^{k} A_i \sigma^{\perp} B_i \right) \right).
    \end{align*}
    Furthermore, we extend existing inequalities for positive operators to the class of sector matrices and prove determinant inequalities for the Hadamard product of sector matrices. 

Keywords

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