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Summary: SEVERAL PROBLEMS IN OPERATOR THEORY
WILLIAM ARVESON
1. Introduction
We discuss some problems and conjectures in higher dimensional operator
theory. These all have something to do with the basic problem of developing
an effective Fredholm theory of d-contractions, and completing the index
theorem that was partially established in Theorem B of [Arv02], following up
on [Arv00]. My opinion is that good progress on any one of these problems
will be a significant advance, if not a breakthrough.
Most of what follows is an exposition of the theory of Dirac operators,
Fredholmness, and index from scratch, in a form accessible to anyone with
a good basic knowledge of operators on Hilbert spaces. The conjectures and
problems will be found in Section 4.
2. Dirac Operators in dimension d
Let ¯T = (T1, . . . , Td) be a multioperator of complex dimension d, that
is to say, a d-tuple of mutually commuting bounded operators acting on a
common Hilbert space H. All geometric properties of ¯T are reflected in
properties of its associated Dirac operator, and we begin by recalling the
basic facts about Dirac operators from [Arv02].
The Dirac operator associated with a d-dimensional multioperator ¯T is
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