Differential properties of the numerical range map of pairs of matrices

✍ Scribed by J.A. Hillman; B.R.F. Jefferies; W.J. Ricker; B. Straub

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 988 KB
• Volume: 267
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(97)80055-9

✦ Synopsis

Let A = (A,, A,) be a pair of Hermitian operators in @" and A = A, + iA,. We investigate certain differential properties of the numerical range map nA: r H )) with the aim of better understanding the nature of the numerical range W(A) of A. For example, the joint eigenvalues of A correspond to the stationary points of n* (i.e. points where the derivative nk vanishes). Moreover, points x where rank ni( x) = 2 get mapped by nA into the interior W(A)" of W(A). For n = 2, it turns out that if A, and A, have no common invariant subspace, then the image under n* of the set Z I( A) consisting of those points x with rank n'J x) = 1 is precisely the boundary aW( A) of W(A), and the image under n* of the rank 2 points for nk is precisely W( A)" ; there are no rank 0 points for ni. As a consequence (for n = 2) we have that A, A, = A, A, iff 8,(A) it n,'(JW( A)).