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Index Theory Beyond the Fredholm Case (Paperback, 1st ed. 2022)
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Index Theory Beyond the Fredholm Case (Paperback, 1st ed. 2022)
Series: Lecture Notes in Mathematics, 2323
Expected to ship within 10 - 15 working days
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This book is about extending index theory to some examples where
non-Fredholm operators arise. It focuses on one aspect of the
problem of what replaces the notion of spectral flow and the
Fredholm index when the operators in question have zero in their
essential spectrum. Most work in this topic stems from the
so-called Witten index that is discussed at length here. The new
direction described in these notes is the introduction of `spectral
flow beyond the Fredholm case'. Creating a coherent picture of
numerous investigations and scattered notions of the past 50 years,
this work carefully introduces spectral flow, the Witten index and
the spectral shift function and describes their relationship. After
the introduction, Chapter 2 carefully reviews Double Operator
Integrals, Chapter 3 describes the class of so-called p-relative
trace class perturbations, followed by the construction of Krein's
spectral shift function in Chapter 4. Chapter 5 reviews the
analytic approach to spectral flow, culminating in Chapter 6 in the
main abstract result of the book, namely the so-called principal
trace formula. Chapter 7 completes the work with illustrations of
the main results using explicit computations on two examples: the
Dirac operator in Rd, and a differential operator on an interval.
Throughout, attention is paid to the history of the subject and
earlier references are provided accordingly. The book is aimed at
experts in index theory as well as newcomers to the field.
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