TY - JOUR

T1 - Grothendieck-Lidskiǐ theorem for subspaces of Lp-spaces

AU - Reinov, O.

AU - Latif, Q.

PY - 2013

Y1 - 2013

N2 - 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 ∑∞ k=1 μk(T). 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. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

AB - 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 ∑∞ k=1 μk(T). 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. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

U2 - 10.1002/mana.201100112

DO - 10.1002/mana.201100112

M3 - Article

SP - 279

EP - 282

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 2-3

ER -