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Summary: p-SUMMABLE COMMUTATORS IN DIMENSION d
WILLIAM ARVESON
Abstract. We show that many invariant subspaces M for d-shifts
(S1, . . . , Sd) of finite rank have the property that the orthogonal pro-
jection PM onto M satisfies
PM Sk - SkPM Lp
, 1 k d
for every p > 2d, Lp
denoting the Schatten-von Neumann class of all
compact operators having p-summable singular value lists. In such cases,
the d tuple of operators ¯T = (T1, . . . , Td) obtained by compressing
(S1, . . . , Sd) to M
generates a -algebra whose commutator ideal is
contained in Lp
for every p > d.
It follows that the C
-algebra generated by {T1, . . . , Td} and the iden-
tity is commutative modulo compact operators, the Dirac operator as-
sociated with ¯T is Fredholm, and the index formula for the curvature
invariant is stable under compact perturbations and homotopy for this
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