Oh yes, this is Von Neumann's stuff. Ya, I know this. Okay, so yes, it will be something:
\[
\begin{align}
H\psi = \int_{-\infty}^\infty e^{ixy}\psi(y)\,d\mu\\
H^t\psi = \int_{-\infty}^\infty k(t,y)\psi(y)\,d\mu\\
\end{align}
\]
For some kernel \(k(t,y)\). Usually they use the Fourier transform, especially when we're talking about neutral/unit circle operations.
This is actually closely related to work I did when I was at U of T, I looked a lot at iterating Linear operators on Hilbert spaces (infinite square matrices acting on an infinite vector). The neutral case is by far the most interesting, but also the most difficult.
IT tends to extend pretty naturally from the \(n x n\) case. If you have some square matrix \(A\), and if you take the exponential:
\[
e^{At}x = \sum_{n=0}^\infty A^nx \frac{t^n}{n!}\\
\]
You can take the Mellin transform in specific cases (you need a specific bounded lemma), where then:
\[
\Gamma(1-z) A^{z-1} x = \int_0^\infty e^{At}x t^{-z}\,dt\\
\]
Then in many cases you can re represent this using a Fourier transform or a Laplace transform, just becomes a kind of elaborate change of variables.
\[
\begin{align}
H\psi = \int_{-\infty}^\infty e^{ixy}\psi(y)\,d\mu\\
H^t\psi = \int_{-\infty}^\infty k(t,y)\psi(y)\,d\mu\\
\end{align}
\]
For some kernel \(k(t,y)\). Usually they use the Fourier transform, especially when we're talking about neutral/unit circle operations.
This is actually closely related to work I did when I was at U of T, I looked a lot at iterating Linear operators on Hilbert spaces (infinite square matrices acting on an infinite vector). The neutral case is by far the most interesting, but also the most difficult.
IT tends to extend pretty naturally from the \(n x n\) case. If you have some square matrix \(A\), and if you take the exponential:
\[
e^{At}x = \sum_{n=0}^\infty A^nx \frac{t^n}{n!}\\
\]
You can take the Mellin transform in specific cases (you need a specific bounded lemma), where then:
\[
\Gamma(1-z) A^{z-1} x = \int_0^\infty e^{At}x t^{-z}\,dt\\
\]
Then in many cases you can re represent this using a Fourier transform or a Laplace transform, just becomes a kind of elaborate change of variables.