Proper holomorphic mappings of the spectral unit ball

TL;DR

This paper proves that the spectral unit ball admits only trivial proper holomorphic mappings, extending classical results to a matrix spectral setting and showing rigidity in complex analysis.

Contribution

It establishes an Alexander type theorem for the spectral unit ball, demonstrating the absence of non-trivial proper holomorphic mappings for dimensions n ≥ 2.

Findings

No non-trivial proper holomorphic mappings in spectral unit ball for n ≥ 2

Extension of classical Alexander theorem to spectral matrix domains

Rigidity result in complex analysis of spectral domains

Abstract

We prove an Alexander type theorem for the spectral unit ball $\Omega_n$ showing that there are no non-trivial proper holomorphic mappings in $\Omega_n$, $n\geq 2$.

Equations

$$\sigma (F(A))={\pi }_n(f({\zeta }_1),\dots ,f({\zeta }_n)),\text{ A∈Ωn is such that }\sigma (A)={\pi }_n({\zeta }_1,\dots ,{\zeta }_n).$$

$$\frac{1}{\alpha }{F}^{\prime }(0)(V)\in \sigma (V).$$

$$\sigma (F(\zeta V))={\pi }_n(f(\zeta {\mu }_1),\dots ,f(\zeta {\mu }_n))$$

$$\sigma \left(\frac{F(\zeta V)}{\zeta }\right)={\pi }_n\left(\frac{(f(\zeta {\mu }_1),\dots ,f(\zeta {\mu }_n))}{\zeta }\right).$$

$$\Phi (\text{Cal}{T}_z)\subset \text{Cal}{T}_z,z\in {\mathbb{C}}^n.$$

$$\text{Cal}{T}_0\not\ni \Phi (N+M)=\Phi (N)+\Phi (M)=\Phi (M)\in \text{Cal}{T}_0$$

$$|\zeta ||1+\zeta |\le \rho ((A+\zeta V{)}^2)\le {\rho }^2(A+\zeta V).$$

$$\rho \left(\frac{F(A+\zeta V)}{\zeta }\right)=\max \left\{\frac{|f({\mu }_j(\zeta ))|}{|\zeta |}:j=1,\dots ,n\right\},$$

$$F_{|\text{Cal}{T}_{{\pi }_n({\lambda }_0,\dots ,{\lambda }_0)}}\text{ is injective.}$$

$$\phi :\mathbb{C}\ni \lambda \mapsto f^{-1}(\lambda C)\in \text{Cal}{T}_{{\pi }_n({\zeta }_1,\dots ,{\zeta }_n)}$$

Full text

The following necessary form of proper holomorphic mappings of the spectral ball, which is a simple consequence of the description of the set of proper holomorphic self-mappings of the symmetrized polydisc, will be crucial in our considerations and justifies the introducing of the condition 1 below.

In view of Proposition 2 it is natural that we study below the holomorphic mappings $F:\Omega_{n}\mapsto\Omega_{n}$ such that there is a function $f\in\Cal{O}(\mathbb{D},\mathbb{D})$ with the property

[TABLE]

We start with the following lemma.

[TABLE]

Actually, $\sigma(\zeta V)=\pi_{n}(\zeta\mu)$, $\zeta\in\mathbb{C}$. Consequently,

[TABLE]

for sufficiently small $\zeta\in\mathbb{D}$ and then

[TABLE]

Passing with $\zeta$ to [math] we get that $F^{\prime}(0)(V)\in\Cal{T}_{\pi_{n}(\alpha\mu)}$. Therefore, $\Phi:=\frac{1}{\alpha}F^{\prime}(0):\Cal{M}_{n}\mapsto\Cal{M}_{n}$ is a linear mapping such that

[TABLE]

To finish the proof of the lemma it is sufficient to show that $\Phi$ is a monomorphism. Suppose that it does not hold. Then there is an $N\in\Cal{M}_{n}$, $N\neq 0$ such that $\Phi(N)=0$. Because of 3 we get that $N\in\Cal{T}_{0}$. But then there is an $M\in\Cal{T}_{0}$ such that $N+M\not\in\Cal{T}_{0}$. In particular,

[TABLE]

– contradiction.

∎

[TABLE]

For any $\zeta\in\mathbb{C}$ there are $\mu_{j}(\zeta)\in\mathbb{C}$, $j=1,\ldots,n$ such that $\pi_{n}(\mu_{1}(\zeta),\ldots,\mu_{n}(\zeta))=\sigma(A+\zeta V)$. Then $\sigma(F(A+\zeta V))=\pi_{n}(f(\mu_{1}(\zeta)),\ldots,f(\mu_{n}(\zeta)))$ for $\zeta\in\mathbb{D}$ small. We also know that $\max\{|\mu_{j}(\zeta)|:j=1,\ldots,n\}=\rho(A+\zeta V)\geq\sqrt{|\zeta|}\sqrt{|1+\zeta|}$ and $\rho(A+\zeta V)\to 0$ as $\zeta\to 0$.

Note that $\rho(\frac{F(A+\zeta V)}{\zeta})\to\rho(F^{\prime}(A)(V))$ as $\zeta\to 0$. But on the other hand

[TABLE]

which tends to infinity as $\zeta\to 0$ because $f^{\prime}(0)\neq 0$ – a contradiction. ∎

Note that the results proven so far referred to a larger class of mappings than only proper ones. It is possible that they may have application to the study of more general mappings than only the proper holomorphic ones.

First we show simple results on the geometry of the sets $\Cal{T}_{z}$.

As a simple corollary of the results on the set of non-derogatory matrices in the sets $\Cal{T}_{z}$ we get the following.

At the moment we are ready to move to the proof of our main result.

We claim that for any $\lambda_{0}\in\mathbb{D}$ such that $B^{\prime}(\lambda_{0})\neq 0$ (note that such points exist) the function

[TABLE]

Actually, making use of the automorphisms of $\Omega_{n}$ and the properties of Blaschke products we may assume that $\lambda_{0}=0$, $B(0)=0$ and $B^{\prime}(0)\neq 0$. It follows from Lemma 4 that $F^{-1}(0)\cap\Cal{T}_{0}=\{0\}$. In particular, $F(0)=0$. Now Lemma 3 applies and we get that $F^{\prime}(0)$ is an isomorphism. Consequently, $F$ is locally invertible near [math]. Note that there is a neighborhood $\Cal{V}$ of [math] such that $\#F^{-1}(A)\cap\Cal{T}_{0}=1$ for any $A\in\Cal{V}\cap\Cal{T}_{0}$. Otherwise there would exist $\Cal{T}_{0}\owns A^{\nu},\tilde{A}^{\nu}$ such that $A^{\nu}\neq\tilde{A}^{\nu}$ and $F(A^{\nu})=F(\tilde{A}^{\nu})\to 0$. But the properness of $F$ implies that (taking if necessary a subsequence) either both sequences $(A^{\nu}),(\tilde{A}^{\nu})$ converge to [math] or at least one of the sequences converges to a non-zero element $\tilde{A}$ from $\Cal{T}_{0}$ such that $F(\tilde{A})=0$. In the first case we contradict the local invertibility of $F$ near [math] and in the second case we get two points in $F^{-1}(0)\cap\Cal{T}_{0}$ – a contradiction, too.

Now the analyticity of the set $\{A\in\Cal{T}_{0}:\#F^{-1}(A)\cap\Cal{T}_{0}=1\}$ (see e.g. [Łoj~1991], Section V.7.1) (the mapping $F_{|\Cal{T}_{0}}:\Cal{T}_{0}\mapsto\Cal{T}_{0}$ is proper and open) and the fact that $\Cal{T}_{0}$ is a cone shows that the mapping $F_{|\Cal{T}_{0}}:\Cal{T}_{0}\mapsto\Cal{T}_{0}$ is a one-to-one mapping.

Now we prove the following property.

(5) Let $z^{\nu}\to z^{0}\in\mathbb{G}_{n}$, where $z^{\nu}\in\mathbb{G}_{n}$ be such that $F_{|\Cal{T}_{z^{\nu}}}$ is not injective for any $\nu$ then $F_{|\Cal{T}_{z^{0}}}$ is not injective.

Actually, to prove 5 note that because of the properties of proper holomorphic mappings we may assume that there are two sequences of non-derogatory matrices $(A^{\nu})$, $(\tilde{A}^{\nu})$ with $A^{\nu}\neq\tilde{A}^{\nu}$ lying in $\Cal{T}_{z^{\nu}}$, $F(A^{\nu})=F(\tilde{A}^{\nu})$ and tending to matrices $A,\tilde{A}\in\Cal{T}_{z^{0}}$ such that $A$ is non-derogatory and $F_{|\Cal{T}_{z^{0}}}$ is locally invertible in $A$. In the case $A\neq\tilde{A}$ we are done, so assume that $A=\tilde{A}$. The local invertibility of $F_{|\Cal{T}_{z^{0}}}$ near $A$ implies that there is a neighborhood $\Cal{U}$ of $A$ such that $F_{|\Cal{T}_{z^{\nu}}\cap\Cal{U}}$ is invertible for $\nu$ large enough, which contradicts the equality $F(A^{\nu})=F(\tilde{A}^{\nu})$.

We claim that

(6) for any $z\in\mathbb{G}_{n}$ the mapping $F_{|\Cal{T}_{z}}$ is injective.

Put $U:=\{z\in\mathbb{G}_{n}:F_{|\Cal{T}_{z}}\text{ is injective}\}$. The fact that $F_{|\Cal{T}_{0}}$ is injective shows that $U$ is not empty. The property 5 shows that $U$ is open. To see that $U$ is closed in $\mathbb{G}_{n}$ take a sequence $U\owns z^{\nu}\to z\in\mathbb{G}_{n}$. Suppose that $z\not\in U$. Then there are different non-derogatory matrices $A_{1},\ldots,A_{k}\in\Cal{T}_{z}$, $C\in\Cal{T}_{w}$ with $k\geq 2$ such that $F^{-1}(C)\cap\Cal{T}_{z}=\{A_{1},\ldots,A_{k}\}$. We may choose arbitrarily small open connected neighborhoods $\Cal{U}_{1},\ldots,\Cal{U}_{k},\Cal{V}$ of $A_{1},\ldots,A_{k},C$ such that $\Cal{U}_{l}\cap\Cal{U}_{p}=\emptyset$ for $l\neq p$, $\Cal{U}_{j}\cap\Cal{T}_{\tilde{z}}$ is connected, $\Cal{V}\cap\Cal{T}_{\tilde{w}}$ is connected for any $\tilde{z},\tilde{w}\in\mathbb{G}_{n}$, $j=1,\ldots,k$ and $F^{-1}(\Cal{V})\subset\bigcup_{j=1}^{k}\Cal{U}_{j}$. Consequently, for any $\nu$ there are pairwise disjoint sets $F(\Cal{U}_{j}\cap\Cal{T}_{z^{\nu}})\cap\Cal{V}$, $j=1,\ldots,k$ that are open in $\Cal{V}\cap\Cal{T}_{w^{\nu}}$ and that are non-empty for $\nu$ large enough. Now the properness of $F$ shows that for $\Cal{V}$ sufficiently small the sets $F(\Cal{U}_{j}\cap\Cal{T}_{z^{\nu}})$, $j=1,\ldots,k$ cover the whole set $\Cal{V}\cap\Cal{T}_{w^{\nu}}$ for $\nu$ large enough; thus contradicting the connectedness of $\Cal{V}\cap\Cal{T}_{w^{\nu}}$.

Since $\mathbb{G}_{n}$ is connected we get that $U=\mathbb{G}_{n}$, so 6 is satisfied.

Let $m$ denote the degree of $B$. We claim that $m=1$. Suppose that $m\geq 2$. Note that taking instead of $F$ the composition of many $F$’s we may assume that $m\geq n$. There is a point $\zeta_{0}\in\mathbb{D}$ such that $B^{-1}(\zeta_{0})=\{\zeta_{1},\ldots,\zeta_{m}\}$. Composing, if necessary, with automorphisms of $\Omega_{n}$ we may assume that $\zeta_{0}=0$. Recall that $\Cal{T}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})}$ is an $n^{2}-n$ dimensional submanifold. Choose $A\in\Cal{T}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})}$ such that $F(A)=0$. Let $f:=F_{|{\Cal{T}}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})}}:\Cal{T}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})}\mapsto\Cal{T}_{0}$. Then $f$ is a holomorphic bijective mapping. Let us fix a regular point $C$ in $\Cal{T}_{0}$. Then the function

[TABLE]

is holomorphic on $\mathbb{C}\setminus\{0\}$ (the points $\lambda C$, $\lambda\in\mathbb{C}\setminus\{0\}$, are regular in $\Cal{T}_{0}$) and continuous at [math] with $\varphi(0)=A$ (use the properness and injectivity of $f$). Consequently, $\varphi$ is holomorphic on $\mathbb{C}$. Note that $(F\circ\varphi)(\lambda)=\lambda C$, $\lambda\in\mathbb{C}$, so the tangent space to $\Cal{T}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})}$ at $A$ i.e. $T_{A}(\Cal{T}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})})$ is mapped onto $H:=F^{\prime}(A)(T_{A}(\Cal{T}_{\pi_{n}(\zeta_{1},\ldots,\zeta_{n})}))$, which contains the vector $F^{\prime}(A)(\varphi^{\prime}(0))=(F\circ\varphi)^{\prime}(0)=C$. Consequently, $H$ contains all regular points of $\Cal{T}_{0}$, so it contains the whole $\Cal{T}_{0}$, which contains $n^{2}-n+1$ linearly independent vectors contradicting the fact that $H$ is at most $n^{2}-n$ dimensional vector space.

Consequently, we have proven that $\#F^{-1}(C)=1$ for $C\in\Cal{\Omega}_{n}$ showing that $F$ is an automorphism. ∎

Acknowledgment

The author wishes to express his gratitude to Witold Jarnicki for fruitful conversations on the properties of proper holomorphic mappings between analytic sets.