Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.

For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ by $||f||_n = \operatorname{sup}(\{|f(z)| : |z|\leq n\})$.

$\big\langle \operatorname{Ent},+,\cdot,\{||.||_n : n\in \{1,2,3,...\}\} \big\rangle$ is a Frechet space.

For all complex numbers $z_0$ and members $g$ of $\operatorname{Ent}$, the operators $L_1,...,L_4 : \operatorname{Ent} \to \operatorname{Ent}$ defined by

$(i) \quad (L_1(f))(z) = g(z)\cdot f(z)$

$(ii) \quad (L_2(f))(z) = f(g(z))$

$(iii) \quad (L_3(f))(z) = f'(z)$

$(iv) \quad (L_4(f))(z) = \displaystyle\int_0^z f$

are all continuous and linear.

Let $S$ be the set of all functions obtainable by the above.

Let $\mathbf{L}$ be continuous operator algebra on $\operatorname{Ent}$.

Let $T$ be the closure of $S$ as a sub-algebra of $\mathbf{L}$.

Does $\:$ $T = \mathbf{L}$ $\:$ ?

If no, is $T$ dense in $\mathbf{L}$? (uniform operator topology)

If no again, is $T$ dense in $\mathbf{L}$ in some weaker topology?