Weak trace-class operator

<h2 id="definition">Definition</h2>
A <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a> A on an infinite dimensional <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> H is weak trace class if μ(n,A) = O(n−1), where μ(A) is the sequence of <a href="/facts/Singular_value/0QWHnqGG">singular values</a>. In mathematical notation the two-sided <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> of all weak trace-class operators is denoted,

L
          
            1
            ,
            ∞
          
        
        =
        {
        A
        ∈
        K
        (
        H
        )
        :
        μ
        (
        n
        ,
        A
        )
        =
        O
        (
        
          n
          
            −
            1
          
        
        )
        }
        .
      
    
    {\displaystyle L_{1,\infty }=\{A\in K(H):\mu (n,A)=O(n^{-1})\}.}

where 
 
 
 
 K
 (
 H
 )
 
 
 {\displaystyle K(H)}
 
 are the compact operators. The term weak trace-class, or weak-L1, is used because the operator ideal corresponds, in J. W. Calkin's <a href="/facts/Calkin_correspondence/8Lhv2xDg">correspondence</a> between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to the <a href="/facts/Lp_space/gbad0Rv1">weak-l1 sequence space</a>.

<h2 id="properties">Properties</h2>
<ul><li>the weak trace-class operators admit a <a href="/facts/Quasinorm/ledoUdJW">quasi-norm</a> defined by</li></ul>

‖
        A
        
          ‖
          
            w
          
        
        =
        
          sup
          
            n
            ≥
            0
          
        
        (
        1
        +
        n
        )
        μ
        (
        n
        ,
        A
        )
        ,
      
    
    {\displaystyle \|A\|_{w}=\sup _{n\geq 0}(1+n)\mu (n,A),}

making L1,∞ a quasi-Banach operator ideal, that is an ideal that is also a <a href="/facts/Quasi-Banach_space/ledoUdJW">quasi-Banach space</a>.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Lp_space/gbad0Rv1">Lp space</a></li>
<li><a href="/facts/Spectral_triple/jsvcMVGj">Spectral triple</a></li>
<li><a href="/facts/Singular_trace/S4xkdjCI">Singular trace</a></li>
<li><a href="/facts/Dixmier_trace/54j3kWo3">Dixmier trace</a></li></ul>

<ul><li>B. Simon (2005). Trace ideals and their applications. Providence, RI: Amer. Math. Soc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-82-183581-4.</li>
<li>A. Pietsch (1987). Eigenvalues and s-numbers. Cambridge, UK: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-52-132532-5.</li>
<li>A. Connes (1994). <a href="https://archive.org/details/noncommutativege0000conn">Noncommutative geometry</a>. Boston, MA: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-185860-5.</li>
<li>S. Lord, F. A. Sukochev. D. Zanin (2012). <a href="http://www.degruyter.com/view/product/177778">Singular traces: theory and applications</a>. Berlin: De Gruyter. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-026255-1.</li></ul>

Weak trace-class operator open-in-new

Weak trace-class operator