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Spectrum of the Real Part of a Weighted Shift Operator
by Stephen Crowley January 21, 2024 1 Introduction This document presents an overview of the spectrum of the real part of a weighted shift operator and explains why it is equal to the closed interval [-1,1]. The explanation is grounded in the principles of functional analysis and operator theory. 2 Preliminaries 2.1 Weighted Shift Operator A weighted shift operator, denoted as T, is a type of linear operator acting on a Hilbert space H. It shifts the elements of a sequence or a function space and multiplies each element by a corresponding weight. The spectral properties of this operator are significantly influenced by these weights. 2.2 Spectrum of an Operator The spectrum of a linear operator T, denoted by σ(T), comprises the set of complex numbers λ for which T-λ*I is not invertible, where I is the identity operator. The spectrum includes eigenvalues and other values where the operator is not invertible. 2.3 Real Part of an Operator The real part of an operator T, denoted as ℜ(T), is given by 1/2*(T+T^∗), where T^∗ is the adjoint of T. This operator is self-adjoint, even if T itself is not. 3 Spectrum of the Real Part of a Weighted Shift Operator The statement in question is that the spectrum of the real part of a weighted shift operator is the closed interval [-1,1]. This can be understood through the following considerations: • The weights of the shift operator, bounded by 1 in absolute value, limit the operator norms of T and T^∗ to 1. • The real part of T combines T and T^∗ and inherits their spectral characteristics. • Consequently, the spectrum of the real part lies within [-1,1], as the operator norms are bounded and the spectrum of a self-adjoint operator is always within the interval defined by its norm. 4 Conclusion The spectrum of the real part of a weighted shift operator being within the interval [-1,1] is a result derived from the bounded nature of the operator and its adjoint, as well as the spectral theorem for self-adjoint operators. This theorem asserts that a self-adjoint operator on a Hilbert space has a real spectrum, comprising eigenvalues or continuous spectrum intervals, which in this case is the interval [-1,1].[1] Bibliography [1] J. DOMBROWSKI. Spectral properties of real parts of weighted shift operators. Indiana University Mathematics Journal, 29(2):249–259, 1980.
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