It may help to remind you of some facts about hilbert spaces. Every functional \( L : \mathcal{H} \to \mathbb{C} \) has a representation as,
\(
L f = (f,c)_{\mathcal{H}}\,\,\text{for some}\,\,c = c(f) \in \mathcal{H}\\
\)
With that in mind,
\(
L f = \uparrow F |_{z=z_0} = (f,c_{z_0})\,\,\text{for some}\,\,c_{z_0} \in \mathcal{H}\\
\)
So we can represent the operator,
\(
\uparrow F = (Hf, x^{-\overline{z}})_{\mathcal{H}}\\
\uparrow F = (f,c_{z})_{\mathcal{H}}\\
\)
Such that \( c_z \) depends on \( f \); so we should write \( c_z = c_z(f) \) (in the linear case we get to throw away the dependence; c is constant)
And we can create such a chain as \( c_z^n(f) \) such that,
\(
\uparrow^n F = (f,c_z^n)_{\mathcal{H}}\\
\)
Because the uparrow operator sends the space to itself. I'm wondering if this will admit an obvious extension to \( n \mapsto s \in \mathbb{C} \). As, this superfunction operation is well managed in this space; and always sends to this space, we're okay.
The spaces in question, again, are \( \mathbb{S}_\theta \) for \( \theta < \pi/2 \) and \( F \in \mathbb{E}_{\theta} \) for \( \theta < \pi/2 \); with the additional requirement that,
\(
|F(z)| \le M\,\,\text{for}\,\,\Re(z) > 0\\
F : \mathb{R}^+ \to \mathbb{R}^+\,\,\,\text{and}\,\,F' : \mathbb{R}^+ \to \mathbb{R}^+\\
F : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0}\\
\)
Where \( \uparrow \) maps such \( F \) to such \( F \). Additionally, for any two \( F_1,F_2 \) for \( c_1,c_2 > 0 \) we get \( c_1F_1 + c_2F_2 = F \) still belongs to this space. This means we have some linear span data. Honestly Mphlee, I'm thinking adjoints is the way to do this. This is making more and more sense. Especially as you describing the categorical perspective; it just looks like complicated hilbert spaces.
\(
L f = (f,c)_{\mathcal{H}}\,\,\text{for some}\,\,c = c(f) \in \mathcal{H}\\
\)
With that in mind,
\(
L f = \uparrow F |_{z=z_0} = (f,c_{z_0})\,\,\text{for some}\,\,c_{z_0} \in \mathcal{H}\\
\)
So we can represent the operator,
\(
\uparrow F = (Hf, x^{-\overline{z}})_{\mathcal{H}}\\
\uparrow F = (f,c_{z})_{\mathcal{H}}\\
\)
Such that \( c_z \) depends on \( f \); so we should write \( c_z = c_z(f) \) (in the linear case we get to throw away the dependence; c is constant)
And we can create such a chain as \( c_z^n(f) \) such that,
\(
\uparrow^n F = (f,c_z^n)_{\mathcal{H}}\\
\)
Because the uparrow operator sends the space to itself. I'm wondering if this will admit an obvious extension to \( n \mapsto s \in \mathbb{C} \). As, this superfunction operation is well managed in this space; and always sends to this space, we're okay.
The spaces in question, again, are \( \mathbb{S}_\theta \) for \( \theta < \pi/2 \) and \( F \in \mathbb{E}_{\theta} \) for \( \theta < \pi/2 \); with the additional requirement that,
\(
|F(z)| \le M\,\,\text{for}\,\,\Re(z) > 0\\
F : \mathb{R}^+ \to \mathbb{R}^+\,\,\,\text{and}\,\,F' : \mathbb{R}^+ \to \mathbb{R}^+\\
F : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0}\\
\)
Where \( \uparrow \) maps such \( F \) to such \( F \). Additionally, for any two \( F_1,F_2 \) for \( c_1,c_2 > 0 \) we get \( c_1F_1 + c_2F_2 = F \) still belongs to this space. This means we have some linear span data. Honestly Mphlee, I'm thinking adjoints is the way to do this. This is making more and more sense. Especially as you describing the categorical perspective; it just looks like complicated hilbert spaces.