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Grothendieck-Lidskiǐ theorem for subspaces of Lp-spaces

Authors

  • Oleg Reinov,

    Corresponding author
    1. Department of Mathematics and Mechanics, St. Petersburg State University, Saint Petersburg, Russia
    2. Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, Pakistan
    • Department of Mathematics and Mechanics, St. Petersburg State University, Saint Petersburg, Russia
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  • Qaisar Latif

    1. Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, Pakistan
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Abstract

In 1955, A. Grothendieck has shown that if the linear operator T in a Banach subspace of an L-space is 2/3-nuclear then the trace of T is well defined and is equal to the sum of all eigenvalues {μk(T)} of T. Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the S1-operator in L2(ν) with its spectral trace equation image. We show that for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p|, and for every s-nuclear operator T in every subspace of any Lp(ν)-space the trace of T is well defined and equals the sum of all eigenvalues of T. Note that for p = 2 one has s = 1, and for p = ∞ one has s = 2/3.

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