In general, what is the logarithmic derivative of an operator? Specifically, the situation I have in mind is as follows:

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for example, $$E_N(\varphi(s),g)=\varphi(s)(g)+M(s)\varphi(s)(g).$$ The resulting the trace formula involves the logarithmic derivative of $M(s)$. In the classical setting this is sometimes known as the determinant of the scattering matrix $\Phi(s)$, and the logarithmic derivative makes sense.

How should I think of $M'(s)M^{-1}(s)$ in general? In particular, in what way is it more than a formal expression mimicking the logarithmic derivative?

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for example, $$E_N(\varphi(s),g)=\varphi(s)(g)+M(s)\varphi(s)(g).$$ The resulting the trace formula involves the logarithmic derivative of $M(s)$. In the classical setting this is sometimes known as the determinant of the scattering matrix $\Phi(s)$, and the logarithmic derivative makes sense.

How should I think of $M'(s)M^{-1}(s)$ in general? In particular, in what way is it more than a formal expression mimicking the logarithmic derivative?

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for example, $$E_N(\varphi(s),g)=\varphi(s)(g)+M(s)\varphi(s)(g).$$ The resulting the trace formula involves the logarithmic derivative of $M(s)$. In the classical setting this is sometimes known as the determinant of the scattering matrix $\Phi(s)$, and the logarithmic derivative makes sense.

How should I think of $M'(s)M^{-1}(s)$ in general? In particular, in what way is it more than a formal expression?

In general, what is the logarithmic derivative of an operator? Specifically, the situation I have in mind is as follows:

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for example, $$E_N(\varphi(s),g)=\varphi(s)(g)+M(s)\varphi(s)(g).$$ The resulting the trace formula involves the logarithmic derivative of $M(s)$. In the classical setting this is sometimes known as the determinant of the scattering matrix $\Phi(s)$, and the logarithmic derivative makes sense.

How should I think of $M'(s)M^{-1}(s)$ in general? In particular, in what way is it more than a formal expression mimicking the logarithmic derivative?