Is there any known results about differentiability properties of the function $\mathbb f:\mathbb R \to\mathbb R,$ $f(t):=\|A+tB\|_{op}$ where $\|.\|_{op}$ denotes the usual operator norm of the matrices acting on finite dimensional complex Hilbert spaces?
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asked Aug 7, 2019 at 12:19
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3$\begingroup$ note that the operator norm, restricted to diagonal matrices, is just the max norm. $\endgroup$– Pietro MajerCommented Aug 7, 2019 at 14:18
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1$\begingroup$ Maybe this question should be migrated to math.stackexchange.con. $\endgroup$– Deane YangCommented Aug 8, 2019 at 0:30
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$\begingroup$ @DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question $\endgroup$– Yemon ChoiCommented Aug 10, 2019 at 3:41
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1 Answer
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It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| \leq 1$ and $|t|$ for $|t| > 1$.
However, $\|A + tB\|$ is Lipschitz in $t$, so it is differentiable almost everywhere by Rademacher's theorem.
answered Aug 7, 2019 at 13:14
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$\begingroup$ Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc. $\endgroup$ Commented Aug 7, 2019 at 21:10
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3$\begingroup$ Or $A=0$, $B=1$, so $f(t)=|t|$. $\endgroup$ Commented Aug 7, 2019 at 22:06
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$\begingroup$ @SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have. $\endgroup$ Commented Aug 10, 2019 at 3:40
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$\begingroup$ Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them $\endgroup$ Commented Aug 10, 2019 at 3:42
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$\begingroup$ Okay. I understand. I wrongly commented at he place where I should have not. $\endgroup$ Commented Aug 10, 2019 at 4:28