arXiv subjects

Mark E. Mancuso

Publications and source records attributed to Mark E. Mancuso.

4 recordsLinked to original sources

arXiv2022

Noncommutative partially convex rational functions

Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. The notion of xy-convexity, a strengthening of partial convexity arising in connection with BMIs, is also considered. A characterization of xy-convex polynomials is given.

math.FA↗

arXiv2021

Noncommutative functions are weakly algebraic on operatorial polynomial polyhedra

An operatorial polynomial polyhedron is a set of the form $$B_δ(\mathcal{B}(\mathcal{H}))=\{X\in \mathcal{B}(\mathcal{H})^d : \Vertδ(X)\Vert<1\}$$ where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded operators on a separable Hilbert space $\mathcal{H}$, and $δ$ is a matrix of polynomials in $d$ noncommuting variables. These sets appear throughout the literature on noncommutative function theory. While much of what has been written involves matricial polynomial polyhedra, there do exist $δ$ such that the associated $B_δ(\mathcal{B}(\mathcal{H}))$ is non-empty but contains no matrix points. Algebraicity of operatorial noncommutative functions has been established in the case that the domain $B_δ(\mathcal{B}(\mathcal{H}))$ is a balanced set (hence contains the matrix point 0). In this paper, we dispense of such assumptions on the domain and prove that an operatorial noncommutative function on any $B_δ(\mathcal{B}(\mathcal{H}))$ is weakly algebraic in the sense that its value at each operator tuple $Z$ lies in the weak operator topology closure of the unital algebra generated by the coordinates of $Z$.

math.FA↗

arXiv2019

Noncommutative partial convexity via $Γ$-convexity

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $Γ$, a free set is called $Γ$-convex if it closed under isometric conjugation by isometries intertwining $Γ$. We establish an Effros-Winkler Hahn-Banach separation theorem for $Γ$-convex sets; they are delineated by linear pencils in the coordinates of $Γ$ and the variables $x$.

math.FA↗

arXiv2019

Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains

We introduce a notion of a noncommutative function defined on a domain of $d$-tuples of bounded operators on an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to merely injectivity.

math.FA↗