Abstract
Mapping theorems for the numerical range analogous to the spectral mapping theorem are hard to come by. The analogy is rather limited by the convexity of the numerical range. However, significant results were obtained in relating the numerical ranges and the numerical radii of an operator T to those of the operator f(T), where f is is a given function. As can be expected, the best results were obtained in the special case f(T) = T n, n a natural number. In addition to the preceding results, this chapter also gives some other results for the numerical range of products, commuting operators, and the natural connections between the numerical range and the theory of dilations of operators.
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Notes and References
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Gustafson, K.E., Rao, D.K.M. (1997). Mapping Theorems. In: Numerical Range. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8498-4_2
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DOI: https://doi.org/10.1007/978-1-4613-8498-4_2
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