Page 40 - Textos de Matemática Vol. 44

30 S. W. DRURY
2. The Result
Before stating our main result, we will need some additional notation. Let D˜ be the convex hull of D ∪ {0}. Then since C \ L is star-shaped at 0, we see that D˜ ∩ L = ∅. Since L is closed and D˜ is compact in C, the separation distance δ between L and D˜ is strictly positive.
Theorem 2.1. Let H be a hermitian matrix with its spectrum in [−1, 1], let Z be a matrix with |||Z − zI||| ≤ r, then we assert that σmin(I − ZH) ≥ δ where σmin denotes the smallest singular value or equivalently the resolvent inequality
holds.
|||(I − ZH)−1||| ≤ δ−1
-2.0 -1.0
2.0
1.0
0.0 0.0
-1.0
1.0
2.0 -2.0
-1.0
2.0
1.0
0.0
0.0 1.0 2.0
-1.0
4.0
3.0
2.0
1.0
0.0
0.0 1.0
Figure 1. Cases 1, 2 and 3.
In the theorem |||
normal, but if it were, then the condition |||Z − zI||| ≤ r would be equivalent to Z having its spectrum contained in D. There are actually three specific cases that can occur.
• Incase1,0∈DandthereforeD˜=D.Thenδ=min|1±z|−r.
• In case 2, 0 ∈/ D, but the closest point of D˜ to L is actually in D. Again
δ = min |1 ± z| − r.
• In case 3, 0 ∈/ D, and the closest point of D˜ to L lies on a straight line
part of the boundary of D˜.
Proof of Theorem 2.1. The result is obvious in the one-dimensional case or if r = 0. We will prove the result under the additional assumption that Re z ̸= 0. The general result follows from this case using continuity. Let Z = zI + rX
||| denotes the operator norm. We do not insist that Z be
-2.0 -1.0
2.0 3.0