Definition 6.2.13: Exponential and Logarithmic

Definition 6.2.13 (Exponential and Logarithmic).label Let $A$ be a unital Banach algebra for each $x\in A$. The exponential

\begin{align*}\exp:\com\to \com&&\lam=a+bi\mto \exp(a+bi)=e^{a}(\cos b+i\sin b)=\sums{\infty}{n=0}\frac{(a+bi)^{n}}{n!}\end{align*}

is an entire function, we define

\begin{align*}\exp x=\sums{\infty}{n=0}\frac{x^{n}}{n!}\end{align*}

which is an absolutely convergent power series where $x^{0}=1$. If $x,y\in A$ commute then

\begin{align*}\exp(x+y)=(\exp x)(\exp y)\end{align*}

In particular $\fall x\in A$

\begin{align*}\exp(-x)=(\exp x)\inv\end{align*}

hence

\begin{align*}\exp:A\to G(A)\end{align*}

Furthermore $\fall t_{0},t,s\in\R$

\begin{align*}\exp(s+t)x=(\exp sx)(\exp tx)&&\limit{t\to t_0}\norm{\exp tx-\exp t_0x}{}=0\end{align*}

thus $\curl{\exp tx:t\in\R}$ is a continuous one parameter subgroup of $G(A)$. The map

\begin{align*}[0,1]\to G(A)&&t\mto \exp tx\end{align*}

is a path joining $1$ to $\exp x$ hence $\exp x$ lies in the connected component $G_{0}(A)$ of $G(A)$ containing the identity. $G_{0}(A)$ (which is closed) is called the principal component of $G(A)$. By 6.1.17, $G(A)$ is open in the Banach space $A$ so it is locally path-connected so $G_{0}(A)$ is a clopen subgroup of $G(A)$. Since $x\in G(A)\mto axa\inv \in G(A)$ is a group automorphism of $G(A)$ for each $a\in G(A)$ which leaves the identity fixed, $G_{0}(A)$ is a normal subgroup of $G(A)$. The quotient group $G(A)/G_{0}(A)$ is called the index group of $A$.

If $\spec{A}{x}$ is contained in the domain of the principal logarithm $\text{Log}:\com\del(-\infty,0]\to\com$ then $\log x\define\text{Log}x$ by 6.2.10. If $\spec{A}{x}\suf \text{dom(Log)}$ then by 6.2.11 we have

\begin{align*}\spec{A}{\log x}=\text{Log}\parens{\spec{A}{x}}&&\exp\log x=x\end{align*}

since $\exp$ is entire. If $\norm{x-1}{sp}<1$ i.e. $\spec{A}{x}\suf\curl{\lam\in \com:\abs{\lam-1}<1}\suf \com\del(-\infty,0]$ then $\log x$ can be written as the absolutely convergent power series

\begin{align*}\log x=-\sums{\infty}{n=1}\frac{(1-x)^{n}}{n}\end{align*}

The Eigenspace

Direct References

The Eigenspace

Direct References