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Mathematical Notes, 2010
The main purpose of this note is to show that the question posed in the paper [1] of D. P. Sinha ... more The main purpose of this note is to show that the question posed in the paper [1] of D. P. Sinha D. P. and A. K. Karn (see the very end of that paper) has a negative answer, and that the answer could have been obtained, essentially, in 1985 after the papers [2], [3] by the author appeared in 1982 and 1985, respectively.
In 1955, A. Grothendieck has shown that if the linear operator T in a Banach subspace of an L ∞ -... more 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 {µ k (T )} of T. V.B. Lidskiǐ , in 1959, proved his famous theorem on the coincidence of the trace of the S 1 -operator in L 2 (ν) with its spectral trace ∞ k=1 µ k (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 L p (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.
Mathematical Notes, 2010
The main purpose of this note is to show that the question posed in the paper [1] of D. P. Sinha ... more The main purpose of this note is to show that the question posed in the paper [1] of D. P. Sinha D. P. and A. K. Karn (see the very end of that paper) has a negative answer, and that the answer could have been obtained, essentially, in 1985 after the papers [2], [3] by the author appeared in 1982 and 1985, respectively.
In 1955, A. Grothendieck has shown that if the linear operator T in a Banach subspace of an L ∞ -... more 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 {µ k (T )} of T. V.B. Lidskiǐ , in 1959, proved his famous theorem on the coincidence of the trace of the S 1 -operator in L 2 (ν) with its spectral trace ∞ k=1 µ k (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 L p (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.