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Maybe an even more elementary argument than the one of Tobias: The continuity of all involved operators is easy: simply all differential operators with smooth coefficients between sections of vector b …

To get an example: consider non-negative smooth functions $\chi_n$ with compact support in the intervals $[0, 1/n]$ with maximum $1$ at some point in this interval. Note that such functions indeed exi …

Just to give an example on how weird this can become: take $E = \mathbb{R}$ and $F = \ell^2$ with standard Hilbert basis $e_0 e_1, e_2, \ldots$. Then take a smooth bump function $\chi \in C^\infty(\ma …

OK, on request, the following reference as answer: Pietsch, A.: Nuclear locally convex spaces, vol. 66 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York, Heidelberg, 1972 …

Let's try this. I didn't check all the estimates but the idea should be roughly as follows: since $M$ is compact, the flow $\Phi$ of $X$ is complete giving a one-parameter group action of $\mathbb{R}$ …

Being a bit late for the party, here is nevertheless a small answer. In fact, there is a rather explicit way to compute adjoints of every differential operator (any order) between (smooth, compactly s …

Suppose for simplicity that $A$ and $B$ are unital. If now $\pi = \pi_1 \otimes \pi_2$ then the operators $\pi(a \otimes 1)$ commute with all the operators of the form $id \otimes B$ with $B \in B(H_2 …

The surprising fact is that the GNSStinespringKasparov theory is in fact completely algebraic, at least to a very large extend: the following results have been obtained by a PhD of mine but are, unfor …

Yes, an $n$ positive map is also $n-1$ positive. Hence you map is $n$-positive for all $n$, i.e. completely positive. For a proof, you include $M_{n-1}(A)$ as upper left block into $M_n(A)$.

Let me give you a counter-example with an associative $\beta$, i.e. a Frechet algebra for which the exponentials do not exist in general: The main reason is that a Frechet algebra needs not to be loca …

Maybe not in a single theorem, but you can go for Cor1 in Section 33 and Cor3 in Section 50 in Treves book.

Well, that depends on what topology you want to put on the space of distributions. The weak$^*$ is probably not really the one you would like to take. Instead, the strong dual might be more useful. Th …

If $\mathcal{A}$ is a complete locally convex (Hausdorff) associative unital algebra (over $\mathbb{C}$) one is interested in defining "transcendental" functions of a given algebra element $a \in \mat …

There is the book by Kriegl and Michor called "Convenient setting of global analysis" published by the AMS. It goes much beyond Fréchet and really gives a big panorama. However, it is not easy reading …

Your question also makes sense for a Banach space and that is where the following results also apply. The Hilbert space case is then only a particular case where you can get some more information abou …