Convergence of the compositional inverse and the noncommutative Jacobian conjecture in Popescu's theory of free biholomorphic functions on $B(\mathcal{H})^n$

This task concerns the two questions left open by G. Popescu in Free biholomorphic functions and operator model theory (J. Funct. Anal. 262 (2012) 3240-3308), and the objective is to resolve them. His free inverse function theorem (Theorem 1.4) assumes that the compositional inverse $g$ of an $n$-tuple $f$ of noncommutative power series with nonzero radius of convergence, $f(0)=0$ and $\det J_f(0)\neq 0$, itself has nonzero radius of convergence; Question 1 asks whether this assumption is automatic. Question 2, the noncommutative Jacobian conjecture in the operator setting, asks whether an $n$-tuple $p$ of noncommutative polynomials whose Jacobian matrix $J_p(X)$ is an invertible operator for every $X\in B(\mathcal{H})^n$ - or only for $X=0$ - induces a polynomial automorphism of $B(\mathcal{H})^n$. The Problem Statement formulates both questions precisely.

Background

In commutative analysis, a convergent power series map $F=(F_1,\dots,F_n)$ in $n$ complex variables with $F(0)=0$ and invertible Jacobian matrix at the origin automatically has a convergent compositional inverse: this is the classical inverse function theorem, and no separate convergence assumption on the inverse is ever needed. The polynomial counterpart of local invertibility is the classical Jacobian conjecture: if a polynomial map $\mathbb{C}^n\to\mathbb{C}^n$ has constant nonzero Jacobian determinant, must it have a polynomial inverse? This question has been open since 1939 (its history and algebraic approaches are surveyed by Belov, Bokut, Rowen and Yu, cited below). The Ax-Grothendieck theorem, stating that an injective polynomial self-map of $\mathbb{C}^n$ is bijective, is a related landmark in the same circle of ideas.

Free (noncommutative) function theory replaces the commuting coordinates $z_1,\dots,z_n$ by freely noncommuting indeterminates $Z_1,\dots,Z_n$. A noncommutative formal power series is a formal sum $\sum_\alpha a_\alpha Z^\alpha$ indexed by the words $\alpha$ in the free semigroup $F_n^+$ generated by $Z_1,\dots,Z_n$ (including the empty word); it is evaluated at a tuple $X=(X_1,\dots,X_n)$ of bounded linear operators on a Hilbert space $\mathcal{H}$ - here $B(\mathcal{H})$ denotes the algebra of all bounded operators on $\mathcal{H}$ - by sending a word $Z_{i_1}\cdots Z_{i_k}$ to the product $X_{i_1}\cdots X_{i_k}$. Following Popescu, an $n$-tuple $f=(f_1,\dots,f_n)$ of such series has nonzero radius of convergence when the series converge uniformly on the operator balls

$[B(\mathcal{H})^n]_r=\{X\in B(\mathcal{H})^n:\; X_1X_1^*+\cdots+X_nX_n^*<r^2I\}$

for some $r>0$ and every Hilbert space $\mathcal{H}$; such tuples are called free holomorphic functions, and they are compatible with direct sums and similarities of operator tuples.

Popescu developed a differential calculus for these objects. The free partial derivative $\partial/\partial Z_i$ is the derivation on noncommutative polynomials and power series determined by the Leibniz rule together with $\partial Z_j/\partial Z_i=\delta_{ij}$ (equivalently, applied to a word it sums over all ways of deleting one occurrence of $Z_i$); the Jacobian matrix of $f$ is $J_f=(\partial f_i/\partial Z_j)_{i,j}$. His Theorem 1.3 shows that an $n$-tuple $f$ with $f(0)=0$ possesses a compositional inverse $g=(g_1,\dots,g_n)$ as a formal power series tuple, meaning $f\circ g=g\circ f=\mathrm{id}$ as formal series, if and only if $\det J_f(0)\neq 0$ - a purely formal, algebraic statement. His Theorem 1.4, the free inverse function theorem, then assumes in addition that both $f$ and $g$ have nonzero radius of convergence and concludes that $f$ is a free biholomorphism between neighborhoods of the origin in $B(\mathcal{H})^n$. The assumption that $g$ converges is not cosmetic: the paper's entire operator model theory - the noncommutative domains $B_f(\mathcal{H})=\{X\in B(\mathcal{H})^n:\; g(f(X))=X,\ \Vert f(X)\Vert\le 1\}$, the universal models $(M_{Z_1},\dots,M_{Z_n})$ of left multiplication by the coordinate functions on the Hardy-type space $\mathcal{H}^2(f)$ of formal power series, and the associated domain algebra, Hardy algebra and $C^*$-algebra - is constructed for tuples in this class.

The paper leaves two questions open. The first asks whether the hypothesis on $g$ in Theorem 1.4 is redundant: if $f$ has nonzero radius of convergence, $f(0)=0$, and $\det J_f(0)\neq 0$, must the formal inverse $g$ also have nonzero radius of convergence? In the commutative case this is automatic, so a negative answer would be a genuinely noncommutative phenomenon. The second question is the Jacobian conjecture in this noncommutative setting: if $p=(p_1,\dots,p_n)$ is an $n$-tuple of noncommutative polynomials such that $J_p(X)$ is an invertible operator for every $X\in B(\mathcal{H})^n$ - or only for $X=0$ - must the map $\Phi_p:X\mapsto(p_1(X),\dots,p_n(X))$ be a polynomial automorphism of $B(\mathcal{H})^n$, that is, must $p$ admit a compositional inverse given by a noncommutative polynomial tuple? Popescu verifies a large class of examples satisfying the global hypothesis (invertible linear tuples modified by triangular polynomial terms, his Proposition 2.3) and notes that the question is trivial when every $p_i$ has degree one. The parenthetical origin-only variant is not an analog of the classical Jacobian conjecture - the Keller hypothesis is a constant nonzero Jacobian determinant, that is, invertibility of the Jacobian at every point rather than at a single point - and it in fact has a negative answer: for $n=1$ and $p(Z)=Z+Z^2$ the word-deletion Jacobian satisfies $J_p(0)=I$, which is invertible, while the unique formal compositional inverse of $p$ (which exists by Popescu's Theorem 1.3) is the Catalan series $Z-Z^2+2Z^3-5Z^4+\cdots$, not a polynomial; a polynomial $q$ with $q\circ p=\mathrm{id}$ would coincide with that unique formal inverse, so no polynomial tuple $q$ satisfies $p(q(X))=X=q(p(X))$ and $\Phi_p$ is not a polynomial automorphism of $B(\mathcal{H})^n$.

Later literature resolves part of the second question, but in a different setting. In matricial free analysis - where free polynomials are evaluated on tuples of $N\times N$ complex matrices of every size $N$, on sets closed under direct sums and similarities, and differentiation is the free (fully matricial Frechet) derivative $Dp(x)[h]$ - Pascoe proved that for a free polynomial map the nonsingularity of $Dp(X)$ at every point is equivalent to injectivity, to bijectivity, and to the inverse agreeing with a free polynomial on each matrix size; this settles the all-points variant of the Jacobian conjecture in the matricial setting and yields a free Ax-Grothendieck theorem. Augat's free Grothendieck theorem then upgraded this: an injective free polynomial map has a genuine free polynomial inverse, with an explicit degree bound $\deg(q)\le(3^n\prod_{i=1}^n i^3)(\deg(p)-1)+1$ and a terminating algorithm that either produces the polynomial inverse or reports non-injectivity. These results do not cover Popescu's setting: they concern finite-dimensional matrix tuples with the free Frechet derivative, not operator tuples in $B(\mathcal{H})^n$ for arbitrary (in particular infinite-dimensional) Hilbert spaces with the word-deletion Jacobian defined above, and they give no information when invertibility is assumed only at the origin. Inverse and implicit function theorems for free noncommutative functions were proved by Abduvalieva and Kaliuzhnyi-Verbovetskyi on uniformly open matricial domains - sets of tuples of matrices of all sizes over operator spaces, closed under direct sums and similarities - where their inverse function theorem assumes that the given map is uniformly analytic (a condition Popescu's tuples satisfy after restriction to matrices of finite size) and that its free difference-quotient derivative (the noncommutative analog of the Frechet derivative, defined by the limit of a difference quotient in a matrix direction) at the base point is invertible, concluding that the compositional inverse is again a uniformly analytic free noncommutative function on a uniformly open matricial neighborhood of the image point; this too is a statement about matrices of finite size, and it does not yield convergence on the operator balls $[B(\mathcal{H})^n]_r$ for every Hilbert space $\mathcal{H}$, including infinite-dimensional ones, that Popescu's radius of convergence demands. Mancuso later proved inverse and implicit function theorems on operator domains $\Omega\subset B(\mathcal{H})^d$ in infinite dimension, the setting closest to Popescu's, but under hypotheses on the derivative (bounded below on the domain, or injective with suitable continuity) that are stronger than, and not comparable to, pointwise invertibility of the evaluated Jacobian. Finally, the purely algebraic Jacobian conjecture for free associative algebras - if the Jacobian matrix of Fox derivatives of an endomorphism of the free algebra $\mathbb{C}\langle t_1,\dots,t_n\rangle$ is invertible (the Fox derivative is a derivation taking values in the tensor square of the free algebra rather than in the free algebra itself), then the endomorphism is an automorphism - was proved by Dicks and Lewin (two generators, 1982) and by Schofield in general, and is surveyed by Belov, Bokut, Rowen and Yu. That algebraic statement concerns invertibility over the free algebra and automorphisms of the free algebra; it is distinct from Popescu's operator-theoretic question about invertibility of evaluated Jacobians as operators and polynomial automorphisms of $B(\mathcal{H})^n$, so its resolution does not close the questions posed here.

Problem Statement

Let $Z_1,\dots,Z_n$ be freely noncommuting indeterminates, let $f=(f_1,\dots,f_n)$ be an $n$-tuple of formal power series in these indeterminates (with complex coefficients) such that $f(0)=0$ and $\det J_f(0)\neq 0$, where $J_f=(\partial f_i/\partial Z_j)_{i,j}$ is the Jacobian matrix for the free partial derivatives determined by the Leibniz rule and $\partial Z_j/\partial Z_i=\delta_{ij}$, and let $g=(g_1,\dots,g_n)$ denote the compositional inverse of $f$ as a formal power series tuple, which exists by Popescu's Theorem 1.3.

Question 1 (radius of convergence of the inverse). If $f$ has nonzero radius of convergence - that is, $f$ is a free holomorphic $n$-tuple on some operator ball $[B(\mathcal{H})^n]_r$ for all Hilbert spaces $\mathcal{H}$ - must $g$ also have nonzero radius of convergence? Equivalently: is the hypothesis on $g$ in Popescu's free inverse function theorem (Theorem 1.4) redundant?

Question 2 (noncommutative Jacobian conjecture on operator tuples). Let $p=(p_1,\dots,p_n)$ be an $n$-tuple of noncommutative polynomials, and let $\Phi_p:B(\mathcal{H})^n\to B(\mathcal{H})^n$ be the induced map $\Phi_p(X)=(p_1(X),\dots,p_n(X))$ for every Hilbert space $\mathcal{H}$. Suppose that the Jacobian matrix $J_p(X)=(\partial p_i/\partial Z_j)(X)$ is an invertible operator matrix either (a) for every $X\in B(\mathcal{H})^n$, or (b) only for $X=0$. Does it follow that $\Phi_p$ is a polynomial automorphism of $B(\mathcal{H})^n$, that is, that there exists an $n$-tuple $q$ of noncommutative polynomials with $p(q(X))=X=q(p(X))$ for all $X\in B(\mathcal{H})^n$ and all Hilbert spaces $\mathcal{H}$?

Question 1 and variant (a) of Question 2 are two independent components, each of which is genuinely open at the operator level stated here, and resolving one of them does not settle the other. Variant (b) of Question 2 is already settled in the Background above, which records its negative answer, so it is not part of the open questions.

References

1. Gelu Popescu, Free biholomorphic functions and operator model theory, Journal of Functional Analysis 262 (2012), 3240-3308. DOI: 10.1016/j.jfa.2012.01.012 (arXiv:1111.5282; the arXiv listing metadata carries the variant title 'Free holomorphic functions and operator model theory', while the posted PDF and the published article are titled as here). https://doi.org/10.1016/j.jfa.2012.01.012
2. J. E. Pascoe, The inverse function theorem and the Jacobian conjecture for free analysis, Mathematische Zeitschrift 278 (2014), 987-994. DOI: 10.1007/s00209-014-1342-2 (arXiv:1303.6011, posted under the preprint title 'The inverse function theorem and the resolution of the Jacobian conjecture in free analysis'). https://doi.org/10.1007/s00209-014-1342-2
3. Meric L. Augat, The free Grothendieck theorem, Proceedings of the London Mathematical Society 118 (2019), no. 4, 787-825. DOI: 10.1112/plms.12200 (arXiv:1712.03929). https://doi.org/10.1112/plms.12200
4. Gulnara Abduvalieva and Dmitry S. Kaliuzhnyi-Verbovetskyi, Implicit/inverse function theorems for free noncommutative functions, Journal of Functional Analysis 269 (2015), 2813-2844. DOI: 10.1016/j.jfa.2015.07.011 (arXiv:1502.05254). https://doi.org/10.1016/j.jfa.2015.07.011
5. Mark E. Mancuso, Inverse and implicit function theorems for noncommutative functions on operator domains, Journal of Operator Theory 83 (2020), no. 2, 447-473. DOI: 10.7900/jot.2018oct21.2237 (arXiv:1804.01040). https://doi.org/10.7900/jot.2018oct21.2237
6. Jim Agler and John E. McCarthy, The implicit function theorem and free algebraic sets, arXiv:1404.6032 (2014). https://arxiv.org/abs/1404.6032
7. Alexei Belov, Leonid Bokut, Louis Rowen and Jie-Tai Yu, The Jacobian Conjecture, together with Specht and Burnside-type problems, in: Automorphisms in Birational and Affine Geometry, Springer Proceedings in Mathematics and Statistics (2014), 249-285. DOI: 10.1007/978-3-319-05681-4_15 (arXiv:1308.0674). https://doi.org/10.1007/978-3-319-05681-4_15
8. W. Dicks and J. Lewin, A Jacobian conjecture for free associative algebras, Communications in Algebra 10 (1982), no. 12, 1285-1306. DOI: 10.1080/00927878208822776 (zbMATH: 3775755). https://doi.org/10.1080/00927878208822776
9. Gelu Popescu, Free biholomorphic functions and operator model theory, II, Journal of Functional Analysis 265 (2013), 786-836. DOI: 10.1016/j.jfa.2013.05.011 (arXiv:1112.0199). https://doi.org/10.1016/j.jfa.2013.05.011