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Some convexity theorems for matrices

Published online by Cambridge University Press:  18 May 2009

P. A. Fillmore  and
J. P. Williams
P. A. Fillmore
Affiliation:
Indiana University, Bloomington, Indiana, U.S.A.
J. P. Williams
Affiliation:
Indiana University, Bloomington, Indiana, U.S.A.
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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 ranges

for k = 1, 2, 3, …. It is clear that W1(A) = W(A). C. A. Berger [2] has shown that Wk(A) is convex.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 2 , September 1971 , pp. 110 - 117
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

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