I've been recently studing Daugavet equation in $L^1[0,1]$ and $C[0,1]$. I understand most of the results I've found but I can't figure out why is it important to find operators that hold Daugavets equation: $$\|I+T\|=1+\|T\|$$
2026-03-28 23:11:43.1774739503
Why's Daugavet equation important?
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The Daugavet equation is of consequence to the philosophy of linear approximation.
Let $E$ be any finite dimensional subspace of $C([0,1])$. Then it follows that there cannot exist any non-zero operator $T:C[0,1]\to E$ such that $T(f)$ is a good approximation of $f$ for all $f\in C[0,1]$. Indeed, because of $\Vert I-T\Vert=1+\Vert T\Vert>1$ we find $f_0\in C[0,1]$ such that $\Vert f_0-T(f_0)\Vert>\Vert f_0\Vert$.
Taken from History of Banach spaces and linear operators. A. Pietsch paragraph 6.9.4.18.