TY - JOUR
T1 - The trace formula in Banach spaces
AU - Johnson, W. B.
AU - Szankowski, A.
N1 - Publisher Copyright:
© 2014, Hebrew University of Jerusalem.
PY - 2014/10
Y1 - 2014/10
N2 - A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalues provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988, Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace formula. A natural class of asymptotically Hilbertian spaces, including some spaces that are ℓ2 sums of finite-dimensional spaces, are Γ-spaces. One consequence is that the direct sum of two Γ-spaces need not be a Γ-space.
AB - A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalues provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988, Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace formula. A natural class of asymptotically Hilbertian spaces, including some spaces that are ℓ2 sums of finite-dimensional spaces, are Γ-spaces. One consequence is that the direct sum of two Γ-spaces need not be a Γ-space.
UR - http://www.scopus.com/inward/record.url?scp=84939889923&partnerID=8YFLogxK
U2 - 10.1007/s11856-014-1107-y
DO - 10.1007/s11856-014-1107-y
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AN - SCOPUS:84939889923
SN - 0021-2172
VL - 203
SP - 389
EP - 404
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -