# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Inequalities for exponentials in Banach algebras

### Tom 100 / 1991

Studia Mathematica 100 (1991), 87-94 DOI: 10.4064/sm-100-1-87-94

#### Streszczenie

For commuting elements x, y of a unital Banach algebra ℬ it is clear that $∥e^{x+y}∥ ≤ ∥e^x∥ ∥e^y∥$. On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form $∥e'^{}∥ ≤ c(1 + |ξ|⟩^s$ for all $ξ ∈ R^m$, where $x = (x_1,..., x_m) ∈ ℬ^m$ and c, s are constants.

• A. J. Pryde

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