Spectral Theory

Lars Tuset
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Lars Tuset: Oslo Metropolitan University, Department of Computer Science

Chapter Chapter 5 in Analysis and Quantum Groups, 2022, pp 131-168 from Springer

Abstract: Abstract In this chapter we develop functional calculi of operators. Given a bounded operator T on a Hilbert space V and a holomorphic function f with a power series that converges absolutely on a disc with radius larger than ∥T∥. Replacing the variable in the power series by T we evidently get a convergent series in the Banach algebra B(V ) producing an operator denoted f(T). The map f↦f(T) is called the holomorphic functional calculus at T. This way complex function theory is brought into the game. In the appendix we study holomorphic calculus for elements in general Banach algebras. There we also include basic complex function theory in the more general context of Banach space valued functions defined on domains in the complex plane, and perform integration of such functions, producing the holomorphic functional calculus as a special case.

Date: 2022
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DOI: 10.1007/978-3-031-07246-8_5

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