In the context of iteration of functions I look at the eigenvalues of the associated matrixoperator/** Carleman-matrix** .

If a function $\small f(x)$ has a negative eigenvalue in its associated carleman-matrix, then the definition of a half-iterate must handle the squareroot of that negative number (and fractional iteration in general). If I create the function $\small g(x)$ by taking the absolute value of that eigenvalue, then $\small f(f(x)) = g(g(x))$ , and moreover $\small f(g(x)) = g(f(x)) $ . (I also assume, that the commutativity makes the solution unique). Then I can do fractional iteration on $ \small g(x) $ and I'm interested in the general relation of that two functions.

Is there a name for that functional relation (for instance "g is the dual of f" or something else so I have a keyword for search) and/or some study online available?

*[update 2]*There was some discussion related to this, where I also was involved answering: "do complex iterates (..) have any meaning?" Here the change-of-sign of the schröder-value, which means an imaginary iteration "height", resembles in some way the change-of-sign of the eigenvalue in the way I looked at in the current question. However, I thought there might be some wider discussion (and possibly a common keyword) to that specific case and the indicated function with "complementary"(?) eigenvalue here.

*[update 1]*After the remark of Gerald Edgar it might be instructive to show at least one example. Motivated by some other question regarding the inversion of $ \small \zeta$ I constructed a carleman-matrix for the zeta-function. Recentering at one fixpoint ( $ \small \sim -0.29590...$ this matrix could be made triangular (call it ZT) and the basic eigenvalue $\small \lambda_1 \sim -0.51273 $ and the others the sequence $\small \lambda_1^0,\lambda_1, \lambda_1^2,\lambda_1^3, \ldots $ Then using diagonalization I replaced the set of eigenvalues by their absolute values, which can also be understood as taking the matrix-squareroot of the square of the matrix ZT, call it ZTA. The entries in ZTA give the coefficients for the function $\small g(x)$ ; the radius of convergence is about $ \small -1.3 < x <0.7$ In this range we find that $\zeta(\zeta(x))=g(g(x))$ at least with visible accuracy..

Here is a picture .

The range nearer to $\small 1-$ must be determined by other means; the twofold iteration of the zeta in this area becomes chaotic and can be seen by the wolfram-alpha call