From bfa5aee60effe4e349c141862bd76189a37e73d6 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 31 May 2026 23:12:15 -0400 Subject: [PATCH] Added the holomorphic functional calculus. --- refs.bib | 10 ++++ src/dg/complex/runge.tex | 16 ++++-- src/op/banach/definitions.tex | 10 ++++ src/op/banach/fc.tex | 96 +++++++++++++++++++++++++++++++++++ src/op/banach/index.tex | 3 +- src/op/banach/spectrum.tex | 16 ++++++ 6 files changed, 146 insertions(+), 5 deletions(-) create mode 100644 src/op/banach/fc.tex diff --git a/refs.bib b/refs.bib index bde87db..1ed6041 100644 --- a/refs.bib +++ b/refs.bib @@ -141,3 +141,13 @@ year={2016}, publisher={CRC Press} } +@book{Takesaki1, + title={Theory of Operator Algebras I}, + author={Takesaki, M.}, + isbn={9783540422488}, + lccn={79013655}, + series={Encyclopaedia of Mathematical Sciences}, + url={https://books.google.ca/books?id=38QIwQEACAAJ}, + year={2001}, + publisher={Springer Berlin Heidelberg} +} diff --git a/src/dg/complex/runge.tex b/src/dg/complex/runge.tex index 62e4cb0..7422ba6 100644 --- a/src/dg/complex/runge.tex +++ b/src/dg/complex/runge.tex @@ -3,10 +3,17 @@ \begin{proposition} \label{proposition:existence-curves} - Let $K \subset \complex$ be compact and $U \in \cn_\complex(K)$, then there exists closed rectifiable curves $\seqf{\gamma_j}$ such that for any separated locally convex space $E$, $f \in H(U; E)$, and $z_0 \in K$, - \[ - f(z_0) = \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(z)}{z - z_0}dz - \] + Let $K \subset \complex$ be compact and $U \in \cn_\complex(K)$, then there exists closed rectifiable curves $\seqf{\gamma_j}$ such that for any separated locally convex space $E$, + \begin{enumerate} + \item For every $f \in H(U; E)$ and $z_0 \in K$, + \[ + f(z_0) = \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(z)}{z - z_0}dz + \] + \item For every $V \in \cn_\complex(U)$, $f \in H(V; E)$, and $z_0 \in V \setminus U$, + \[ + \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(z)}{z - z_0}dz = 0 + \] + \end{enumerate} \end{proposition} \begin{proof}[Proof, {{\cite[Proposition VIII.1.1]{ConwayComplex}}}. ] Via \hyperref[fattening]{proposition:distance-compact}, let $V \in \cn_\complex^o(K)$ precompact with $\ol V \subset U$. Identify $\complex = \real^2$, then since $V$ is precompact, there exists $\delta > 0$ and $\seqf{(x_j, y_j)} \subset U$ such that: @@ -60,6 +67,7 @@ Finally, let $1 \le j \le m$. Since $\mu_j$ does not form a redundant pair, $\mu_j(1)$ intersects at most two distinct squares by (3). In which case, there must exist $1 \le k \le m$ such that $\mu_k(0) = \mu_j(1)$. Therefore $\seqf[m]{\mu_j}$ forms a collection of closed rectifiable curves. \end{proof} + \begin{lemma} \label{lemma:rational-curve-approximation} Let $\gamma \in C([a, b]; \complex)$ be a rectifiable curve, $K \subset \complex$ such that $K \cap \gamma([a, b]) = \emptyset$, $f \in C(\gamma([a, b]); \complex)$, and $\eps > 0$, then there exists $R \in \complex(z)$ such that: diff --git a/src/op/banach/definitions.tex b/src/op/banach/definitions.tex index 3ef240d..0c6c79c 100644 --- a/src/op/banach/definitions.tex +++ b/src/op/banach/definitions.tex @@ -15,4 +15,14 @@ Let $A$ be a Banach algebra, then $A$ is \textbf{unital} if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, $1$ is the unique \textbf{multiplicative identity} of $A$. \end{definition} +\begin{definition}[Homomorphism] +\label{definition:banach-algebra-homomorphism} + Let $A, B$ be Banach algebras and $\phi: A \to B$, then $\phi$ is a \textbf{homomorphism} if: + \begin{enumerate} + \item $\phi \in L(A; B)$. + \item For each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$. + \end{enumerate} +\end{definition} + + diff --git a/src/op/banach/fc.tex b/src/op/banach/fc.tex new file mode 100644 index 0000000..5bb2472 --- /dev/null +++ b/src/op/banach/fc.tex @@ -0,0 +1,96 @@ +\section{The Holomorphic Functional Calculus} +\label{section:holomorphic-functional-calculus} + + +\begin{definition}[Holomorphic Functional Calculus] +\label{definition:holomorphic-functional-calculus} + Let $A$ be a unital Banach algebra and $x \in A$, then there exists a unique continuous homomorphism + \[ + H(\sigma_A(x); \complex) \to A \quad f \mapsto f(x) + \] + + such that: + \begin{enumerate} + \item $1(x) = 1$. + \item $\text{Id}(x) = x$. + \end{enumerate} + + Moreover, + \begin{enumerate}[start=2] + \item For each $U \in \cn_\complex(\sigma_A(x))$ and closed rectifiable curves $\seqf{\gamma_j}$ on $U \setminus \sigma_A(x)$ such that $f(z_0) = \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} f(z)/(z - z_0)dz$ for all $f \in H(U; \complex)$ and $z_0 \in \sigma_A(x)$, + \[ + f(x) = \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(\lambda)}{\lambda - x}d\lambda + \] + + for all $f \in H(U; \complex)$. + \end{enumerate} + + The mapping $f \mapsto f(x)$ is the \textbf{holomorphic functional calculus} of $x$. +\end{definition} +\begin{proof}[Proof, {{\cite[Proposition I.2.7]{Takesaki1}}}. ] + (Definition): Let $U, V \in \cn_\complex(\sigma_A(x))$ such that $\ol V \subset U$, then by \autoref{proposition:existence-curves}, there exists closed rectifiable curves $\seqf{\gamma_j}$ on $V \setminus \sigma_A(x)$ such that + \begin{enumerate}[label=(\alph*)] + \item For all $f \in H(V; \complex)$ and $z_0 \in \sigma_A(x)$, + \[ + f(z_0) = \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(z)}{z - z_0}dz + \] + + \item For all $f \in H(U; \complex)$ and $z_0 \in U \setminus V$, + \[ + \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(z)}{z - z_0}dz = 0 + \] + \end{enumerate} + + For each $f \in H(U; \complex)$, define + \[ + f(x) = \frac{1}{2\pi i}\sum_{j = 1}^n \int_{\gamma_j} \frac{f(\lambda)}{\lambda - x}d\lambda + \] + + then by \hyperref[Cauchy's Theorem]{theorem:cauchy-homotopy}, the above definition is independent of the choice of curves satisfying (a). + + (Linearity): By \autoref{proposition:rs-bound}, the mapping $f \mapsto f(z)$ is a continuous linear map from $H(\sigma_A(x); \complex)$ to $A$. + + (Homomorphism): Let $T$ be the union of the image of $\seq{\gamma_j}$. By \autoref{proposition:existence-curves}, there exists closed rectifiable curves $\seqf[m]{\mu_j}$ on $U \setminus \ol V$ such that $g(z_0) = \frac{1}{2\pi i}\sum_{j = 1}^m \int_{\mu_j} g(z)/(z - z_0)dz$ for all $f \in H(U; \complex)$ and $z_0 \in \ol V$. Now, + \begin{align*} + f(x)g(x) &= \frac{1}{(2\pi i)^2}\sum_{j = 1}^n \sum_{k = 1}^m \braks{\int_{\gamma_j} \frac{f(z)}{z - x}dz} \cdot \braks{\int_{\mu_k}\frac{g(w)}{w - x}dw} \\ + &= \frac{1}{(2\pi i)^2}\sum_{j = 1}^n \sum_{k = 1}^m \int_{\gamma_j}\int_{\mu_k} \frac{f(z)g(w)}{(z - x)(w - x)}dwdz + \end{align*} + + For each $z, w \in U \setminus \sigma_A(x)$ with $z \ne w$, by the \hyperref[resolvent equation]{lemma:resolvent-equation}, + \[ + \frac{f(z)g(w)}{(z - x)(w - x)} = \frac{f(z)g(w)}{w - z}\braks{\frac{1}{z - x} - \frac{1}{w - x}} + \] + + By assumptions on $\seqf[m]{\mu_j}$, for each $1 \le j \le n$, + \[ + \frac{1}{2\pi i }\int_{\gamma_j} \sum_{k = 1}^m \int_{\mu_k} \frac{f(z)g(w)}{(w - z)(z - x)}dwdz = \int_{\gamma_j} \frac{f(z)g(z)}{z - x}dz + \] + + By assumption (b) and \hyperref[Fubini's Theorem]{theorem:rs-fubini}, for each $1 \le k \le m$, + \[ + \sum_{j = 1}^n\int_{\gamma_j}\int_{\mu_k} \frac{f(z)g(w)}{(w - z)(z - x)}dwdz = 0 + \] + + Therefore + \[ + f(x)g(x) = \frac{1}{2\pi i }\sum_{j = 1}^n \int_{\gamma_j} \frac{f(z)g(z)}{z - x}dz = (fg)(x) + \] + + and the mapping $f \mapsto f(x)$ is a homomorphism. + + (1): Since the constant $1$ function is the identity in $H(\sigma_A(x); \complex)$, $1(x) = 1$ by the homomorphism property. + + (2): Let $R > 0$ such that $\sigma_A(x) \subset B_\complex(0, R)$, then by \autoref{proposition:rs-complete} and \hyperref[Cauchy's Integral Formula]{theorem:cauchy-formula}, + \[ + \text{Id}(x) = \frac{1}{2\pi i }\int_{\omega_{0, R}} \frac{1}{z - x}dz = \frac{1}{2\pi i }\sum_{n = 0}^\infty\int_{\omega_{0, R}} z^{-n-1}x^n dz = 1 + \] + + and + \[ + \text{Id}(x) = \frac{1}{2\pi i }\int_{\omega_{0, R}} \frac{z}{z - x}dz = \frac{1}{2\pi i }\sum_{n = 0}^\infty\int_{\omega_{0, R}} z^{-n}x^n dz = x + \] + + (Uniqueness): By (2), the homomorphism extends uniquely to $\complex(z) \cap H(\sigma_A(x); \complex)$. By \hyperref[Runge's Theorem]{corollary:runge-rational-approximation}, it extends uniquely to $H(\sigma_A(x); \complex)$ by continuity. + +\end{proof} + diff --git a/src/op/banach/index.tex b/src/op/banach/index.tex index 0cf7901..341ac20 100644 --- a/src/op/banach/index.tex +++ b/src/op/banach/index.tex @@ -4,4 +4,5 @@ \input{./definitions.tex} \input{./invertible.tex} \input{./igroup.tex} -\input{./spectrum.tex} \ No newline at end of file +\input{./spectrum.tex} +\input{./fc.tex} diff --git a/src/op/banach/spectrum.tex b/src/op/banach/spectrum.tex index 4b80321..130d7e5 100644 --- a/src/op/banach/spectrum.tex +++ b/src/op/banach/spectrum.tex @@ -34,6 +34,22 @@ By \autoref{proposition:banach-algebra-inverse}, the mapping $y \mapsto y^{-1}$ is smooth on $G(A)$. Hence $R_x: \complex \setminus \sigma_A(x) \to A$ is holomorphic. \end{proof} +\begin{lemma}[Resolvent Equation] +\label{lemma:resolvent-equation} + Let $A$ be a unital Banach algebra, $x \in A$, and $\lambda, \mu \in \complex \setminus \sigma_A(x)$, then + \[ + R_x(\lambda) - R_x(\mu) = (\mu - \lambda)R_x(\lambda) R_x(\mu) + \] +\end{lemma} +\begin{proof} + \begin{align*} + [R_x(\lambda) - R_x(\mu)](\mu - x) &= (\lambda - x)^{-1}(\mu - x) - 1 \\ + (\lambda - x)[R_x(\lambda) - R_x(\mu)](\mu - x)&= (\mu - x) - (\lambda - x) = \mu - \lambda \\ + R_x(\lambda) - R_x(\mu) &= (\mu - \lambda)R_x(\lambda)R_x(\mu) + \end{align*} +\end{proof} + + \begin{proposition} \label{proposition:spectrum-non-empty}