diff --git a/refs.bib b/refs.bib index c472d15..bde87db 100644 --- a/refs.bib +++ b/refs.bib @@ -132,3 +132,12 @@ year={1993}, publisher={Taylor \& Francis} } +@book{FollandHarmonic, + title={A Course in Abstract Harmonic Analysis}, + author={Folland, G.B.}, + isbn={9781498727150}, + series={Textbooks in Mathematics}, + url={https://books.google.ca/books?id=z-GYCgAAQBAJ}, + year={2016}, + publisher={CRC Press} +} diff --git a/src/op/banach/spectrum.tex b/src/op/banach/spectrum.tex index 0da111c..4b80321 100644 --- a/src/op/banach/spectrum.tex +++ b/src/op/banach/spectrum.tex @@ -65,7 +65,7 @@ \label{proposition:spectral-radius-hadamard} Let $A$ be a unital Banach algebra and $x \in A$, then $[\cdot]_{sp} = \limsup_{n \to \infty}\normn{x^n}_A^{1/n}$. \end{proposition} -\begin{proof} +\begin{proof}[Proof, {{\cite[Theorem 1.8]{FollandHarmonic}}}. ] Let $r = \limsup_{n \to \infty}\normn{x^n}_A^{1/n}$, then for any $\lambda \in \complex$ with $|\lambda| > r$, the series $\sum_{n = 0}^\infty \lambda^{-n-1}x^n$ converges absolutely, and to the inverse of $(x - \lambda)$. Therefore $r \ge [\cdot]_{sp}$. Let $D = \bracs{\lambda \in \complex|\ |\lambda| > [x]_{sp}}$, then since the series $\sum_{n = 0}^\infty \lambda^{-n-1}x^n$ is the expansion of $R_x$ at infinity, which is defined on $D$, it must converge on $D$. Let $\phi \in A^*$, then by the preceding discussion, the series $\sum_{n = 0}^\infty \lambda^{-n-1}\dpn{x^n, \phi}{A}$ converges on $D$. Thus for any $\lambda \in D$,