Geodesic
Some convexity theorems for matrices
Glasgow mathematical journal, Tome 12 (1971) no. 2, pp. 110-117

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The numerical range of a bounded linear operator A on a complex Hilbertspace H is the set W(A) = {(Af, f): ║f║ = 1}. Because it is convex andits closure contains the spectrum of A, the numerical range is often a useful toolin operator theory. However, even when H is two-dimensional, the numerical range of an operator can be large relative to its spectrum, so that knowledge of W(A) generally permits only crude information about A. P. R. Halmos [2] has suggested a refinement of the notion of numerical range by introducing the k-numerical rangesfor k = 1, 2, 3, .... It is clear that W1(A) = W(A). C. A. Berger [2] has shown that Wk(A) is convex. 
Fillmore, P. A.; Williams, J. P. Some convexity theorems for matrices. Glasgow mathematical journal, Tome 12 (1971) no. 2, pp. 110-117. doi: 10.1017/S0017089500001221
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