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Given a closed, densely defined linear operator $A : D(A) \subset X \to X$ on a Banach space $X$, there are natural ways to define $f(A)$ as a bounded and linear operator on $X$, $f$ being a suitable holomorphic mapping on some neighbourhood of the spectrum of $A$. On the other hand, in case $A$ is a normal operator on a Hilbert space, it makes sense to consider $f(A)$ even for measurable mappings on the spectrum of $A$.
In the talk, hibrid situations of Banach spaces $X$ and linear operators $A$ which admits coherent versions as non-negative, self-adjopints operators in some linked Hilbert spaces are considered. Then, the possibility of defining $f(A)$ as a bounded operator on $X$, for real differentiable mappings $f : [0,+\infty) \to \mathbb{C}$, is explored. Finally, some resolvent estimates in maximun-norm for the space discretizations of elliptic opertators are presented.
