Riesz spectral projection

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a unital complex Banach algebra, let $a\in A$, and let $E\subseteq {\sigma }_A(a)$ be clopen in the spectrum, that is, both $E$ and ${\sigma }_A(a)\setminus E$ are relatively open in ${\sigma }_A(a)$ (Spectrum and resolvent set in a Banach algebra). Equivalently $E\subseteq {\sigma }_A(a)$ is closed in $\mathbb{C}$ — hence compact — and ${\sigma }_A(a)\setminus E$ is compact as well, and the two are disjoint.

Choose disjoint open sets $U_1\supseteq E$ and $U_0\supseteq {\sigma }_A(a)\setminus E$; such sets exist because $E$ and ${\sigma }_A(a)\setminus E$ are disjoint compact subsets of the plane. Let

${\chi }_E(z):=\{\begin{array}{ll}1,& z\in U_1,\\ 0,& z\in U_0,\end{array}$

a locally constant function on the open neighbourhood $U_1\cup U_0$ of ${\sigma }_A(a)$, hence holomorphic there. The Riesz spectral projection of $a$ associated with $E$ is the calculus value

$P_E:={\chi }_E(a)=\frac{1}{2\pi i}{\int }_{\mathrm{\Gamma }}{\chi }_E(z)R(z,a)dz\in A,$

where $\mathrm{\Gamma }$ is any cycle with trace in $(U_1\cup U_0)\setminus {\sigma }_A(a)$ whose index is $1$ at every point of $E$, whose index is $0$ at every point of ${\sigma }_A(a)\setminus E$, and whose index is $0$ outside $U_1\cup U_0$ — for instance the difference $c_{\mathrm{a}\mathrm{l}\mathrm{l}}-c_E$, where $c_{\mathrm{a}\mathrm{l}\mathrm{l}}$ is admissible for $({\chi }_E,U_1\cup U_0)$ and $c_E$ is a cycle with index $1$ on the compact set ${\sigma }_A(a)\setminus E$ and index $0$ outside $U_0$ (the zero cycle when ${\sigma }_A(a)\setminus E=\mathrm{\varnothing }$): the difference has index $1-0=1$ on $E$, index $1-1=0$ on ${\sigma }_A(a)\setminus E$, and index $0$ outside $U_1\cup U_0$, because $c_{\mathrm{a}\mathrm{l}\mathrm{l}}$ has index $0$ there and $c_E$ has index $0$ outside $U_0\subseteq U_1\cup U_0$. Such cycles exist by Admissible cycle around a compact plane set applied to the two compact sets ${\sigma }_A(a)$ and ${\sigma }_A(a)\setminus E$.

Here the equality with the displayed integral, and its independence of the separating cycle, do not use contour independence outside its admissible-cycle hypothesis. Indeed, put $F(z)={\chi }_E(z)R(z,a)$ on $\mathrm{\Omega }:=(U_1\setminus E)\cup U_0$. This is Banach-valued holomorphic: it is $R(z,a)$ on $U_1\setminus E$ and identically zero on $U_0$, so in particular it extends holomorphically across ${\sigma }_A(a)\setminus E$. If $c_{\mathrm{a}\mathrm{l}\mathrm{l}}$ is admissible and $\mathrm{\Gamma }$ has the separating indices just specified, then $c_{\mathrm{a}\mathrm{l}\mathrm{l}}-\mathrm{\Gamma }$ has index zero on $E$ and outside $U_1\cup U_0$, hence is null-homologous in $\mathrm{\Omega }$. Therefore Banach-valued Cauchy integral vanishes gives ${\int }_{c_{\mathrm{a}\mathrm{l}\mathrm{l}}}F(z)dz={\int }_{\mathrm{\Gamma }}F(z)dz\mathrm{.}$ The left side is the defining calculus integral for ${\chi }_E(a)$. Thus every such $\mathrm{\Gamma }$ gives $P_E$, while germ independence of the calculus makes the value independent of the chosen $U_0,U_1$.

• The function ${\chi }_E$ is a germ, and that is all the definition needs. Its definition depends on the chosen neighbourhoods, but every two such locally constant functions agree on a neighbourhood of ${\sigma }_A(a)$, and the calculus depends only on the germ (Holomorphic functional calculus is contour independent).

• When $E={\sigma }_A(a)$ or $E=\mathrm{\varnothing }$. If $E={\sigma }_A(a)$ then ${\chi }_E=1$ on a neighbourhood of the spectrum and $P_E=1$; if $E=\mathrm{\varnothing }$ then ${\chi }_E=0$ on a neighbourhood of the spectrum and $P_E=0$. Both are consistent with the definition and with the multiplicativity of the calculus (Holomorphic functional calculus homomorphism).

• No idempotence is assumed here. That $P_E^2=P_E$ and that $P_E$ commutes with $a$ are consequences of multiplicativity of the calculus, not part of the definition; they are proved for operators in Riesz spectral projection properties.

• Why the contour has index one on $E$ and zero on the rest of the spectrum. This makes the integral a function of the spectral subset $E$ alone: replacing the cycle by another with the same indices does not change the value, as in the calculus at large. For a single isolated eigenvalue $\lambda$ the projection is the classical residue (ex-riesz-projection-for-a-matrix-with-separated-spectrum).

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