Schur multiplier operator and matrix inequalities

Document Type : Research Paper

Author

Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran.

Abstract

In this note we obtain a reverse version of the Haagerup Theorem. In particular, if $ A \in \mathbb{M}_{n}$ has a $ 2\times2- $ principal submatrix as $ \left[ \begin{array}{cc}
1& \alpha \\
\beta & 1\\
\end{array}
\right]$ with $ \beta \neq \bar{\alpha}, $ then $ \Vert S_{A} \Vert > 1$ where the operator $ S_{A}:\mathbb{M}_{n}\longrightarrow \mathbb{M}_{n} $ is defined by $S_{A}(B) := A \circ B $ where $ "\circ " $ stands for Schur product.

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References

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Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 383-390

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History

  • Receive Date: 22 April 2023
  • Revise Date: 31 August 2023
  • Accept Date: 29 September 2023

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