bo198214 Wrote:But if we translate this into matrix notation by simply replacing \( \circ \) by the matrix multiplication and replacing each function by the corresponding Bell matrix (which is the transposed Carleman matrix) then we see a diagonalization of \( B_b \) because the Bell matrix (and also the Carleman matrix) of \( \mu_{\ln(a)}(x)=\ln(a)x \) is just your diagonal matrix \( {_dV}(\ln(a)) \)!
Hmm, this point disppeared for me, when I read your post first time.
So I could say, that this is already fixed, what I always called "my hypothesis about the set of eigenvalues"? I didn't see something like this in the articles I have access to.
Further it may then be interesting to develop arguments for the degenerate case... In my "increasing size" (of the truncated Bell-matrix) analyses the courious aspect for base \( b= \eta \), appeared, that the set of eigenvalues show decreasing distances between them but also new eigenvalues pop up, and each new eigenvalue was smaller than the previous. See page "Graph"So there are concurring tendencies - and it may be fruitful to do an analysis, why this/how this could be compatible with the/a limit case, where they are assumed to approach 1 asymptotically.
Gottfried
Gottfried Helms, Kassel