Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-03T19:49:12.613Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1.Fillmore, P. A., On similarity and the diagonal of a matrix, Amer. Math. Monthly 76 (1969), 167–169.CrossRefGoogle Scholar
2.Halmos, P. R., A Hilbert Space Problem Book (Princeton, 1967).Google Scholar
3.Horn, Alfred, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620–630.CrossRefGoogle Scholar
4.John, F., On symmetric matrices whose eigenvalues satisfy linear inequalities, Proc. Amer. Math. Soc. 17 (1966), 1140–1146.CrossRefGoogle Scholar
5.Lerer, L. E., On the diagonal elements of normal matrices (Russian), Mat. lssled. 2 (1967), 156–163.Google Scholar
6.MacCluer, C. R., On extreme points of the numerical range of normal operators, Proc. Amer. Math. Soc. 16 (1965), 1183–1184.CrossRefGoogle Scholar
7.Williams, J. P., On compressions of matrices, J. London Math. Soc. 3 (1971), 526–530.CrossRefGoogle Scholar