Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial f f in d d freely noncommuting arguments, find a free polynomial p n p_n , of degree at most n n , to minimize c n ≔ ‖ p n f − 1 ‖ 2 c_n ≔\|p_nf-1\|^2 . (Here the norm is the ℓ 2 \ell ^2 norm on coefficients.) We show that c n → 0 c_n\to 0 if and only if f f is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the d d -shift.

AbstractWe establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

We establish a theory of NC functions on a class of von Neumann algebras with a particular direct sum property, e.g. $B(\mathcal{H})$. In contrast to the theory's origins, we do not rely on appealing to results from the matricial case. We prove that the $k^{\text{th}}$ directional derivative of any NC function at a scalar point is a $k$-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g. the non-commutative polydisk and non-commutative row ball.

We show that the derivative of a noncommutative free analytic map must be free-curl free -- an analog of having zero curl. Moreover, under the assumption that the free domain is connected, this necessary condition is sufficient. Specifically, if $T$ is analytic free demilinear (linear in half its variables) map defined on a connected free domain then $DT(X,H)[K,0] = DT(X,K)[H,0]$ if and only if there exists an analytic free map $f$ such that $Df(X)[H] = T(X,H)$.

Given a tuple $E=(E_1,dots,E_g)$ of $dtimes d$ matrices, the collection of those tuples of matrices $X=(X_1,dots,X_g)$ (of the same size) such that $| sum E_jotimes X_j|le 1$ is called a spectraball $mathcal B_E$. Likewise, given a tuple $B=(B_1,dots,B_g)$ of $etimes e$ matrices the collection of tuples of matrices $X=(X_1,dots,X_g)$ (of the same size) such that $I + sum B_jotimes X_j +sum B_j^* otimes X_j^*succeq 0$ is a free spectrahedron $mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:mathcal B_Eto mathcal D_B$ normalized by $p(0)=0$ and $pu0027(0)=I$ if and only if $mathcal B_E=mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.

We provide an effective single-matrix criterion, in terms of what we call the elementary Pick matrix, for the solvability of the noncommutative Nevanlinna-Pick interpolation problem in the row ball, and provide some applications. In particular we show that the so-called “column-row property” fails for the free semigroup algebras, in stark contrast to the analogous commutative case. Additional applications of the elementary Pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting. Finally we present some numerical results related to the failure of the column-row property.

The main result of this paper establishes the free analog of Grothendieck's theorem on bijective polynomial mappings of Cg. Namely, we show if p is a polynomial mapping in g freely non-commuting variables sending g-tuples of matrices (of the same size) to g-tuples of matrices (of the same size) that is injective, then it has a free polynomial inverse. Other results include an algorithm that tests if a free polynomial mapping p has a polynomial inverse (equivalently is injective; equivalently is bijective). Further, a class of free algebraic functions, called hyporational, lying strictly between the free rational functions and the free algebraic functions are identified. They play a significant role in the proof of the main result.

Given a tuple E=(E_1,…,E_g) of d× d matrices, the collection of those tuples of matrices X=(X_1,…,X_g) (of the same size) such that ∑ E_j⊗ X_j≤ 1 is called a spectraball ℬ_E. Likewise, given a tuple B=(B_1,…,B_g) of e× e matrices the collection of tuples of matrices X=(X_1,…,X_g) (of the same size) such that I + ∑ B_j⊗ X_j +∑ B_j^* ⊗ X_j^*≽ 0 is a free spectrahedron 𝒟_B. Assuming E and B are irreducible, plus an additional mild hypothesis, there is a free bianalytic map p:ℬ_E→𝒟_B normalized by p(0)=0 and p'(0)=I if and only if ℬ_E=ℬ_B and B spans an algebra. Moreover p is unique, rational and has an elegant algebraic representation.

Linear matrix inequalities (LMIs) \(I_d + \sum _{j=1}^g A_jx_j + \sum _{j=1}^g A_j^*x_j^* \succeq 0\) play a role in many areas of applications. The set of solutions of an LMI is a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. Free spectrahedra are exactly the free semialgebraic convex sets. This paper studies free analytic maps between free spectrahedra and, under certain (generically valid) irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in g variables sit in correspondence with g-dimensional algebras. If two bounded free spectrahedra \({\mathcal {D}}_A\) and \({\mathcal {D}}_B\) meeting our irreducibility assumptions are free bianalytic with map denoted p, then p must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a g-dimensional algebra spanned by \((U-I)A_1,\ldots ,(U-I)A_g\) for some unitary U. Furthermore, B and UA are unitarily equivalent. The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from \({\mathcal {D}}_A\) to \({\mathcal {D}}_B\) (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps.

Subsets of the set of $g$-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, contained in the hull of a single point.