Operator monotone function

<h2 id="definition">Definition</h2>
<p class="note">See also: <a href="/facts/Trace_inequality/jtYypUYj">Trace inequality § Operator monotone</a></p>
<p>A function 
  
    
      
        f
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f:I\to \mathbb {R} }
  
 defined on an interval 
  
    
      
        I
        ⊆
        
          R
        
      
    
    {\displaystyle I\subseteq \mathbb {R} }
  
 is said to be operator monotone if whenever 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 are <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrices</a> (of any size/dimensions) whose <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> all belong to the domain of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and whose difference 
  
    
      
        A
        −
        B
      
    
    {\displaystyle A-B}
  
 is a <a href="/facts/Positive_semi-definite_matrix/Hbr6AuPS">positive semi-definite matrix</a>, then necessarily 
  
    
      
        f
        (
        A
        )
        −
        f
        (
        B
        )
        ≥
        0
      
    
    {\displaystyle f(A)-f(B)\geq 0}

where 
  
    
      
        f
        (
        A
        )
      
    
    {\displaystyle f(A)}
  
 and 
  
    
      
        f
        (
        B
        )
      
    
    {\displaystyle f(B)}
  
 are the values of the <a href="/facts/Matrix_function/T2e9sFm9">matrix function</a> <a href="/facts/Matrix_function/T2e9sFm9">induced by 
  
    
      
        f
      
    
    {\displaystyle f}
  
</a> (which are matrices of the same size as 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
).
</p><p>Notation
</p><p>This definition is frequently expressed with the notation that is now defined. 
Write 
  
    
      
        A
        ≥
        0
      
    
    {\displaystyle A\geq 0}
  
 to indicate that a matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is <a href="/facts/Positive_semi-definite_matrix/Hbr6AuPS">positive semi-definite</a> and write 
  
    
      
        A
        ≥
        B
      
    
    {\displaystyle A\geq B}
  
 to indicate that the difference 
  
    
      
        A
        −
        B
      
    
    {\displaystyle A-B}
  
 of two matrices 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 satisfies 
  
    
      
        A
        −
        B
        ≥
        0
      
    
    {\displaystyle A-B\geq 0}
  
 (that is, 
  
    
      
        A
        −
        B
      
    
    {\displaystyle A-B}
  
 is positive semi-definite). 
</p><p>With 
  
    
      
        f
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f:I\to \mathbb {R} }
  
 and 
  
    
      
        A
      
    
    {\displaystyle A}
  
 as in the theorem's statement, the value of the <a href="/facts/Matrix_function/T2e9sFm9">matrix function</a> 
  
    
      
        f
        (
        A
        )
      
    
    {\displaystyle f(A)}
  
 is the matrix (of the same size as 
  
    
      
        A
      
    
    {\displaystyle A}
  
) <a href="/facts/Analytic_function_of_a_matrix/T2e9sFm9">defined in terms of</a> its 
  
    
      
        A
      
    
    {\displaystyle A}
  
's <a href="/facts/Spectral_theorem/Vqc04B21">spectral decomposition</a> 
  
    
      
        A
        =
        
          ∑
          
            j
          
        
        
          λ
          
            j
          
        
        
          P
          
            j
          
        
      
    
    {\displaystyle A=\sum _{j}\lambda _{j}P_{j}}
  
 by

f
        (
        A
        )
        =
        
          ∑
          
            j
          
        
        f
        (
        
          λ
          
            j
          
        
        )
        
          P
          
            j
          
        
         
        ,
      
    
    {\displaystyle f(A)=\sum _{j}f(\lambda _{j})P_{j}~,}
  
 
where the 
  
    
      
        
          λ
          
            j
          
        
      
    
    {\displaystyle \lambda _{j}}
  
 are the eigenvalues of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 with corresponding <a href="/facts/Projection_(linear_algebra)/sElXGkxD">projectors</a> 
  
    
      
        
          P
          
            j
          
        
        .
      
    
    {\displaystyle P_{j}.}
  
 
</p><p>The definition of an operator monotone function may now be restated as:
</p><p>A function 
  
    
      
        f
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f:I\to \mathbb {R} }
  
 defined on an interval 
  
    
      
        I
        ⊆
        
          R
        
      
    
    {\displaystyle I\subseteq \mathbb {R} }
  
 said to be <i>operator monotone</i> if (and only if) for all positive integers 
  
    
      
        n
        ,
      
    
    {\displaystyle n,}
  
 and all 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrices</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 with eigenvalues in 
  
    
      
        I
        ,
      
    
    {\displaystyle I,}
  
 if 
  
    
      
        A
        ≥
        B
      
    
    {\displaystyle A\geq B}
  
 then 
  
    
      
        f
        (
        A
        )
        ≥
        f
        (
        B
        )
        .
      
    
    {\displaystyle f(A)\geq f(B).}

</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Matrix_function/T2e9sFm9">Matrix function</a> – Function that maps matrices to matricesPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Trace_inequality/jtYypUYj">Trace inequality</a> – Concept in Hlibert spaces mathematics</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Schilling, R.; Song, R.; <a href="/facts/Zoran_Vondra%C4%8Dek/klELH434">Vondraček, Z.</a> (2010). <i>Bernstein functions. Theory and Applications</i>. Studies in Mathematics. Vol. 37. de Gruyter, Berlin. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2F9783110215311">10.1515/9783110215311</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783110215311.</li>
<li>Hansen, Frank (2013). "The fast track to Löwner's theorem". <i>Linear Algebra and Its Applications</i>. 438 (11): 4557–4571. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1112.0098">1112.0098</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.laa.2013.01.022">10.1016/j.laa.2013.01.022</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119607318">119607318</a>.</li>
<li>Chansangiam, Pattrawut (2015). <a href="https://doi.org/10.1155%2F2015%2F649839">"A Survey on Operator Monotonicity, Operator Convexity, and Operator Means"</a>. <i>International Journal of Analysis</i>. 2015: 1–8. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1155%2F2015%2F649839">10.1155/2015/649839</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Löwner, K.T. (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift. 38: 177–216. doi:10.1007/BF01170633. S2CID 121439134. <a href="http://eudml.org/doc/168495" target="_blank">http://eudml.org/doc/168495</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Löwner–Heinz inequality". Encyclopedia of Mathematics. <a href="https://www.encyclopediaofmath.org/index.php/Löwner–Heinz_inequality" target="_blank">https://www.encyclopediaofmath.org/index.php/Löwner–Heinz_inequality</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Chansangiam, Pattrawut (2013). "Operator Monotone Functions: Characterizations and Integral Representations". arXiv:1305.2471 [math.FA]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Operator monotone function open-in-new

Operator monotone function