- For $n \times n$ unitary operator $A$, $dist(z, \sigma(A)) = \lVert (z - A)^{-1}\rVert^{-1}$?
- Assume B is a diagonal matrix with two terms are complex number $a$ and $b$, the rest are 1. Do we have $dist(z, \sigma(AB)) \leq C \lVert (z - AB)^{-1}\rVert^{-1}$ for some constant $C$? Thanks!
$dist(z, \sigma(A))$ for $n \times n$ unitary operator $A$
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functional-analysis
spectral-theory
1 Answers
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For any normal operator $M$, we have $\|M\| = \rho(M) = \max \{|\lambda|:\lambda \in \sigma(M)\}$.
For 1, we note that $\sigma(zI - A) = \{|(z - \lambda)^{-1}|:\lambda \in \sigma(A)\}$. We then have $$ \|\sigma(zI - A)^{-1}\| = \max \{|z- \lambda|^{-1}\} = [\min\{|z-\lambda|\}]^{-1} $$ Thus, we have $$ \|\sigma(zI - A)^{-1}\| = \min\{|z-\lambda|\} = dist(z,\sigma(A)) $$ Not sure about 2. However, my guess is "probably not". I'd suggest $2 \times 2$ numerical experiments to see.