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How far is \({\left( 1 + \frac{a}{n}\right) ^n} \) from \( e^a \) when a is an element of a Banach algebra?

  • Vito Lampret
  • Philip G. Spain
Article
  • 26 Downloads

Abstract

We present effective upper and lower bounds for the distance from \(\displaystyle \left( 1 + \frac{a}{n}\right) ^n \) to \(\displaystyle e^a \) for an element a of a complex unital Banach algebra and positive integer n. Specifically
$$\begin{aligned} \frac{1}{2n} \sup \left\{ \left| \mathfrak {R}(z^2) \right| e^{\mathfrak {R}(z)} : z \in \sigma (a) \right\} \lesssim _{(2)} \left\| e^a - \left( 1 + \frac{a}{n}\right) ^n \right\| \le \frac{ \left\| a \right\| ^2}{2n} \ e^{ \left\| a \right\| }, \end{aligned}$$
where \(\sigma (a)\) is the spectrum of a. The symbol \(\lesssim _{(p)}\) means “less than or equal to, up to a term of order \(n^{-p}\)”as discussed below. Following some technical preliminaries (Sect. 1) we treat the real case (Sect.  2), extend to the complex case (Sect.  3), and then generalise to the case of a norm-unital Banach algebra (Sect.  4).

Keywords

Asymptotic approximation Banach algebra Distance Estimate Exponential function Hermitian element Inequalities Norm Spectrum 

Mathematics Subject Classification

Primary: 40A25 41A99 46H99 Secondary: 26D99 47A50 65D20 

Notes

Acknowledgements

The authors are glad to take this opportunity to thank the referee for valuable comments and remarks which have enhanced the quality of the paper.

References

  1. 1.
    Bonsall, F.F., Duncan, J.: Complete normed algebras. Springer, Berlin (1973)CrossRefGoogle Scholar
  2. 2.
    Lampret, V.: Approximating the powers with large exponents and bases close to unit, and the associated sequence of nested limits. Int. J. Contemp. Math. Sci. 6, 2135–2145 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of LjubljanaLjubljanaSlovenia
  2. 2.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

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