Some convexity theorems for matrices
- 1 September 1971
- journal article
- research article
- Published by Cambridge University Press (CUP) in Glasgow Mathematical Journal
- Vol. 12 (2) , 110-117
- https://doi.org/10.1017/s0017089500001221
Abstract
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.Keywords
This publication has 3 references indexed in Scilit:
- On Similarity and the Diagonal of a MatrixThe American Mathematical Monthly, 1969
- On Symmetric Matrices Whose Eigenvalues Satisfy Linear InequalitiesProceedings of the American Mathematical Society, 1966
- On Extreme Points of the Numerical Range of Normal OperatorsProceedings of the American Mathematical Society, 1965