Notes on the numerical range


 by

Joel H. Shapiro

 

Not for Publication
 Abstract: This is an introduction to the notion of numerical range for bounded linear operators on Hilbert space. The main results are: determination of the numerical range for two by two matrices, the Toeplitz-Hausdorff Theorem establishing the convexity of the numerical range for any Hilbert space operator, and a detailed discussion of the relationship between the numerical range and the spectrum. The high points of this latter topic are: containment of the spectrum in the closure of the numerical range, and Hildebrandt's theorem which asserts that the intersection of the closures of the numerical ranges of all operators similar to a given one T is the precisely the convex hull of the spectrum of T.

 

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