Slightly different, but easier to make explicit: For every $t\in[0,1]$ put $K(t,0)=0$, and if $s$ belongs to $I_n=(1/(n+1),1/n]$ put $K(t,s)=\max\{n^2(n+1)(1-\lvert nt-1\rvert),0\}$. Then on the one hand $\int_{1/(n+1)}^{1/n}K(1/n,s)ds=\int_{I_n}n^2(n+1)ds=n\to\infty$. On the other hand, for fixed $t$ the condition $K(t,s)>0$ holds only if $s\in I_n$ with $\lvert nt-1\rvert<1$. This holds only for finitely many $n$ so that $K(t,\cdot)$ is a simple function.

As I mentioned, a sufficient condition is the equi-integrability of the family $\{K(t,\cdot):t\in I\}$, that is $\sup_{t\in I}\lVert\chi_{E_n}(\cdot)K(t,\cdot)\rVert_{L_1(I)}\to0$ whenever $E_1\supseteq E_2\supseteq\ldots$ are measurable with $\bigcap_nE_n=\emptyset$. An integrable majorant $h$ (that is, $\lvert K(t,s)\rvert\le h(s)$ for almost all $s$), is sufficient for this (Lebesgue's dominated convergence theorem) but not necessary.

$x+y^2$ for $0\le y\le1$, $x+y$ otherwise. Please note that this forum is for research-level questions, only. Homework is on-topic on math.stackexchange.com

There are at least 3 definitions of "essential spectrum" which are not equivalent (though the corresponding essential spectral radius is the same). Which definition do you mean?

An infinite-dimensional normed space is a famous example where no natural approach is known. In practice, all "usable" measures on such spaces have the property that most sets have infinite measure. If you want to have a ball with a finite measure in such a space, the measure becomes very unnatural (not translation invariant, extremely weighted, most sets have measure 0, etc.).

I never heard the name Fréchet in connection of compactness criteria in spaces of measurable functions. It might be that he has proven some other result using similar ideas for the proof or that he has found the result independently and that Brezis therefore might want to honor him. But if Brezis does not mention this is hard to verify. Besides that, I agree with Leo Moos that the naming conventions in a country often prefer Mathematician of the same country. (For instance, the Banach–Caccioppoli fixed point theorem is outside of Italy usually only attributed to Banach.)

Concerning holomorphic functional calculus for vector-valued holomorphic functions: While there are several useful notions of a spectrum for nonlinear operators (see e.g. J. Appell, E. De Pascale, A. Vignoli, Nonlinear spectral theory), there is no hope to obtain a corresponding functional calculus due to lack of commutativity and distributivity of nonlinear operators: Already the 3 representations of the same quadratic polynomial x^2+x=x(x+1)=(x+1)x give 2 different results even for simple nonlinear operators/maps.