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, xn,ynn is an element of N subset of XxY\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left( x_{n},y_{n}\right) _{n\in \mathbb {N}}\subset X\times Y$$\end{document} convergent sequences with limn ->infinity xn=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }x_{n}=x$$\end{document}, limn ->infinity yn=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lim \nolimits _{n\rightarrow \infty }y_{n}=y$$\end{document} and T:XxY -> Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:X\times Y\rightarrow Z$$\end{document} a continuous bilinear operator with T1,1=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\left( \textbf{1},\textbf{1}\right) = \textbf{1}$$\end{document}. Then for all sequences of natural numbers ann is an element of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( a_{n}\right) _{n\in \mathbb {N}}$$\end{document} with limn ->infinity an=infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }a_{n}=\infty $$\end{document} we have limn ->infinity T & prod;k=1nexkank+nk+2n,& prod;k=1neykank+2nk+3nan=eln43Tx,1+ln2T1,y;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\left[ T\left( \prod \limits _{k=1}<^>{n}e<^>{ \frac{x_{k}}{a_{n}\left( k+n\right) \left( k+2n\right) }},\prod \limits _{k=1}<^>{n}e<^>{\frac{y_{k}}{a_{n}\left( k+2n\right) \left( k+3n\right) } }\right) \right] <^>{a_{n}}=e<^>{\left( \ln \frac{4}{3}\right) T\left( x,\textbf{ 1}\right) +\left( \ln 2\right) T\left( \textbf{1},y\right) }; \end{aligned}$$\end{document}limn ->infinity T & prod;k=1ncosxkannn+k,& prod;k=1ncoskyknann2+k2nan2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\left[ T\left( \prod \limits _{k=1}<^>{n}\cos \frac{x_{k}}{a_{n}\sqrt{n\left( n+k\right) }},\prod \limits _{k=1}<^>{n}\cos \frac{ky_{k}}{na_{n}\sqrt{n<^>{2}+k<^>{2}}}\right) \right] <^>{na_{n}<^>{2}} \end{aligned}$$\end{document}=e-ln2Tx2,1+1-pi 4T1,y22.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} =e<^>{-\frac{\left( \ln 2\right) T\left( x<^>{2},\textbf{1}\right) +\left( 1- \frac{\pi }{4}\right) T\left( \textbf{1},y<^>{2}\right) }{2}}. \end{aligned}$$\end{document}