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Norm inequalities for accretive-dissipative block matrices


Abstract

Let $ T=[T_{ij}]\in \mathbb{M} _{mn}(\mathbb{C})$ be accretive-dissipative, where $T_{ij}\in \mathbb{M} _{n}(\mathbb{C} )$ for $i,j=1,2,...,m.$ Let $f$ be a function that is convex and increasing on $ [0,\infty )$ where $f(0)=0.$ Then $$ \left\vert \left\vert \left\vert f\left(\sum_{i < j}\left\vert T_{ij}\right\vert^{2}\right) +f\left(\sum_{i < j}\left\vert T_{ji}^{\ast}\right\vert^{2}\right) \right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert f\left( \frac{m^{2}-m}{2}\left\vert T\right\vert^{2}\right) \right\vert \right\vert \right\vert. $$ Also, if $f$ is concave and increasing on $[0,\infty )$ where $f(0)=0$, then% \begin{equation*} \left\vert \left\vert \left\vert f\left( \sum\limits_{i < j}\left\vert T_{ij}\right\vert ^{2}\right) +f\left( \sum\limits_{i < j}\left\vert T_{ji}^{\ast }\right\vert ^{2}\right) \right\vert \right\vert \right\vert \leq (2m^{2}-2m)\left\vert \left\vert \left\vert f\left( \frac{\left\vert T\right\vert ^{2}}{4}\right) \right\vert \right\vert \right\vert. \end{equation*}

Нехай $T=T_{ij}\in \mathbb{M}_{mn}(\mathbb{C} )$, де $T_{ij}\in \mathbb{M}_{n}(\mathbb {C})$ при $i,j=1,2,...,m.$, - акретивно-дисипативна матриця. Нехай $f$ - опукла функція, яка зростає на $ [0,\infty )$, де $f(0)=0.$ Тоді \begin{equation*} \left\vert \left\vert \left\vert f\left( \sum\limits_{i < j}\left\vert T_{ij}\right\vert ^{2}\right) +f\left( \sum\limits_{i < j}\left\vert T_{ji}^{\ast }\right\vert ^{2}\right) \right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert f\left( \frac{m^{2}-m}{2}\left\vert T\right\vert ^{2}\right) \right\vert \right\vert \right\vert. \end{equation*} Також, якщо $f$ є угнутою, зростає на $[0,\infty )$ і $f(0)=0$, то \begin{equation*} \left\vert \left\vert \left\vert f\left( \sum\limits_{i < j}\left\vert T_{ij}\right\vert ^{2}\right) +f\left( \sum\limits_{i < j}\left\vert T_{ji}^{\ast }\right\vert ^{2}\right) \right\vert \right\vert \right\vert \leq (2m^{2}-2m)\left\vert \left\vert \left\vert f\left( \frac{\left\vert T\right\vert ^{2}}{4}\right) \right\vert \right\vert \right\vert. \end{equation*}

Key words: Accretive-dissipative matrix; convex function; concave function; inequality; singular value; unitarily invariant norm.


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Article Information

TitleNorm inequalities for accretive-dissipative block matrices
SourceMethods Funct. Anal. Topology, Vol. 26 (2020), no. 3, 201-215
DOI10.31392/MFAT-npu26_3.2020.02
MilestonesReceived 15.07.2020; Revised 29.07.2020
CopyrightThe Author(s) 2020 (CC BY-SA)

Authors Information

Fadi Alrimawi
Department of Basic Sciences, Al-Ahliyyah Amman University, Amman, Jordan

Mohammad Al-Khlyleh
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan

Fuad A. Abushaheen
Basic Science Department, Middle East University, Amman, Jordan


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Fadi Alrimawi, Mohammad Al-Khlyleh, and Fuad A. Abushaheen, Norm inequalities for accretive-dissipative block matrices, Methods Funct. Anal. Topology 26 (2020), no. 3, 201-215.


BibTex

@article {MFAT1393,
    AUTHOR = {Fadi Alrimawi and Mohammad Al-Khlyleh and Fuad A. Abushaheen},
     TITLE = {Norm inequalities for accretive-dissipative block matrices},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {26},
      YEAR = {2020},
    NUMBER = {3},
     PAGES = {201-215},
      ISSN = {1029-3531},
       DOI = {10.31392/MFAT-npu26_3.2020.02},
       URL = {http://mfat.imath.kiev.ua/article/?id=1393},
}


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