ALMOST MULTIPLICATIVE MAPS AND $\epsilon$ SPECTRUM OF AN ELEMENT IN FRECHET Q-ALGEBRA

Abstract

Let $(A,(p_{k}))$ be a Fr$\acute{\mathbf{\text{e}}}$chet $Q$-algebra with unit $e_{A}$. The $\varepsilon-$ spectrum of an element $x$ in $A$ is defined by,
\begin{center}
$\sigma_{\varepsilon}(x)=\lbrace \lambda \in \mathbb{C}:p_{k_{0}}(\lambda e_{A} -x)p_{k_{0}}(\lambda e_{A} -x)^ {-1}\geq {\frac{1}{\varepsilon}}\rbrace$
\end{center}
for $0<\varepsilon<1$.
We show that there is a close relation between
the $\varepsilon-$spectrum and almost multiplicative maps.
In some situations, it is also shown that
\begin{center}
$ \sigma_{\varepsilon}(x)=\lbrace \varphi(x): \varphi \in M_{alm}(A) , \varphi(e_{A})=1 \rbrace $
\end{center}
for every $x \in A$, where
$M_{alm}(A)$ is the set of all almost multiplicative maps from $A$ to $\mathbb{C}$.