New limits of the Lie product formula type in Banach algebras

We find new limits of the Lie product formula type in Banach algebras with unit. Some sample results: Let X, Y, Z be Banach algebras with unit, \( \left( x_{n},y_{n}\right) _{n\in \mathbb {N}}\subset X\times Y\) convergent sequences with \(\lim \nolimits _{n\rightarrow \infty }x_{n}=x\), \( \lim \nolimits _{n\rightarrow \infty }y_{n}=y\) and \(T:X\times Y\rightarrow Z\) a continuous bilinear operator with \(T\left( \textbf{1},\textbf{1}\right) = \textbf{1}\). Then for all sequences of natural numbers \(\left( a_{n}\right) _{n\in \mathbb {N}}\) with \(\lim \nolimits _{n\rightarrow \infty }a_{n}=\infty \) we have

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\left[ T\left( \prod \limits _{k=1}^{n}e^{ \frac{x_{k}}{a_{n}\left( k+n\right) \left( k+2n\right) }},\prod \limits _{k=1}^{n}e^{\frac{y_{k}}{a_{n}\left( k+2n\right) \left( k+3n\right) } }\right) \right] ^{a_{n}}=e^{\left( \ln \frac{4}{3}\right) T\left( x,\textbf{ 1}\right) +\left( \ln 2\right) T\left( \textbf{1},y\right) }; \end{aligned}$$

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\left[ T\left( \prod \limits _{k=1}^{n}\cos \frac{x_{k}}{a_{n}\sqrt{n\left( n+k\right) }},\prod \limits _{k=1}^{n}\cos \frac{ky_{k}}{na_{n}\sqrt{n^{2}+k^{2}}}\right) \right] ^{na_{n}^{2}} \end{aligned}$$

$$\begin{aligned} =e^{-\frac{\left( \ln 2\right) T\left( x^{2},\textbf{1}\right) +\left( 1- \frac{\pi }{4}\right) T\left( \textbf{1},y^{2}\right) }{2}}. \end{aligned}$$