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We consider pairs of anti-commuting $2p$-by-$2p$ Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of $2$-by-$2$ blocks on which the first matrix has eigenvalues $\pm x_j$ and the second has eigenvalues $\pm y_j$. We call $\{ (x_j, y_j) \}$ the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling.

We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random variable that takes values in the set of commuting $d$-tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the Hermitian case, we characterize the eigenvalue distribution as $n$ tends to infinity. In the non-Hermitian case, we get a formula that holds if the set is irreducible. We show that there are qualitative differences between the single matrix case and the several commuting matrices case.

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $\phi\not\in\mathsf{Aut}(\mathbb D)$, find a non-constant inner function $\Psi$ satisfying the functional equation $\Psi\circ\phi=\tau\Psi$, where $\tau$ is a unimodular constant. We prove that this problem has a solution if and only if $\phi$ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $\tau=1$. If in addition, $\phi$ is parabolic, we prove that the problem has a solution $\Psi$ for $any$ unimodular $\tau$. Finally, we show that if $\phi$ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $\tau$ satisfy $f\circ\phi=\tau f$.

I share my opinions on how to get the most out of conferences. I have some advice on giving talks, but most of the essay is devoted to the much larger share of the conference not spent giving talks: attending talks and socializing.

We prove that the invariant subspaces of the Hardy operator on $L^2[0,1]$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.

We study the algebra $\mathcal{A}$ generated by the Hardy operator $H$ and the operator $M_x$ of multiplication by $x$ on $L^2[0,1]$. We call $\mathcal{A}$ the Hardy-Weyl algebra. We show that its quotient by the compact operators is isomorphic to the algebra of functions that are continuous on $\Lambda$ and analytic on the interior of $\Lambda$ for a planar set $\Lambda$ = $[-1,0] \cup \bar{ \mathbb{D}(1,1)}$, which we call the lollipop. We find a Toeplitz-like short exact sequence for the $C^*$-algebra generated by $\mathcal{A}$. We study the operator $Z = H - M_x$, show that its point spectrum is $(-1,0] \cup \mathbb{D}(1,1)$, and that the eigenvalues grow in multiplicity as the points move to $0$ from the left.

We study monomial operators on $ L^2[0,1]$, that is bounded linear operators that map each monomial $x^n$ to a multiple of $x^{p_n}$ for some $p_n$. We show that they are all unitarily equivalent to weighted composition operators on a Hardy space. We characterize what sequences $p_n$ can arise. In the case that $p_n$ is a fixed translation of $n$, we give a criterion for boundedness of the operator.

The Hardy operator has all the monomial functions as eigenvectors. We study bounded operators on L^2 that take monomial functions to multiples of other monomials, with a shifted exponent. We prove that they all leave the space of functions vanishing on [0,s] invariant. We prove an asymptotic Muntz-Szasz theorem, characterizing the set of functions that are limits of linear combinations of monomials with exponents between n and 2n.

We show that for every connected analytic subvariety $V$ there is a pseudoconvex set $\Omega$ such that every bounded matrix-valued holomorphic function on $V$ extends isometrically to $\Omega$. We prove that if $V$ is two analytic disks intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to $\Omega$, then so does every matrix-valued function. In the special case that $\Omega$ is the symmetrized bidisk, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=\varphi g$, where $g$ is cyclic, $\varphi$ is a contractive multiplier, and $\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization., Comment: 63 pages

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., Comment: 21 pages

We characterize the common range of the adjoints of cyclic multiplication operators on the Drury--Arveson space. We show that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. To achieve this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the strictly smaller Smirnov class of the Drury-Arveson space, and that this in turn equals the common range of the adjoints of cyclic multiplication operators.

We prove a variant of the well-known result that intertwiners for the bilateral shift on `$\ell^2(Z)$ are unitarily equivalent to multiplication operators on $L^2(T)$. This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts., Comment: 25 pages, 7 figures

We compare the quantitative risk of infection from short range airborne transmission of SARS-CoV-2 to the long-range risk from aerosol transmission in an enclosed space.

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space., Comment: 40 pages; minor changes

We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.

A calcular algebra is a subalgebra of $H^\infty(\Omega)$ with norm given by $\| \phi \| = \sup \| \phi(T) \|$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $\O$. We discuss what algebras arise this way, and how they can be represented.

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables., Comment: 75 pages, 1 figure. Appeared in Trans. London Math. Soc.5, no. 1, [2018] 132:183. Updated post to add an extra hypothesis in the statement of Prop. 3.4. (This error appears in the published version.)

For $s\in \mathbb R$ the weighted Besov space on the unit ball $\mathbb B_d$ of $\mathbb C^d$ is defined by $B^s_\omega=\{f\in \operatorname{Hol}(\mathbb B_d): \int_{\mathbb B_d}|R^sf|^2 \omega dV<\infty\}.$ Here $R^s$ is a power of the radial derivative operator $R= \sum_{i=1}^d z_i\frac{\partial}{\partial z_i}$, $V$ denotes Lebesgue measure, and $\omega$ is a radial weight function not supported on any ball of radius $< 1$. Our results imply that for all such weights $\omega$ and $\nu$, every bounded column multiplication operator $B^s_\omega \to B^t_\nu \otimes \ell^2$ induces a bounded row multiplier $B^s_\omega \otimes \ell^2 \to B^t_\nu$. Furthermore we show that if a weight $\omega$ satisfies that for some $\alpha >-1$ the ratio $\omega(z)/(1-|z|^2)^\alpha$ is nondecreasing for $t_0<|z|<1$, then $B^s_\omega$ is a complete Pick space, whenever $s\ge (\alpha+d)/2$., Comment: 32 pages

Let $\mathcal H$ be the Drury-Arveson or Dirichlet space of the unit ball of $\mathbb C^d$. The weak product $\mathcal H\odot\mathcal H$ of $\mathcal H$ is the collection of all functions $h$ that can be written as $h=\sum_{n=1}^\infty f_n g_n$, where $\sum_{n=1}^\infty \|f_n\|\|g_n\|<\infty$. We show that $\mathcal H\odot\mathcal H$ is contained in the Smirnov class of $\mathcal H$, i.e. every function in $\mathcal H\odot\mathcal H$ is a quotient of two multipliers of $\mathcal H$, where the function in the denominator can be chosen to be cyclic in $\mathcal H$. As a consequence we show that the map $\mathcal N \to clos_{\mathcal H\odot\mathcal H} \mathcal N$ establishes a 1-1 and onto correspondence between the multiplier invariant subspaces of $\mathcal H$ and of $\mathcal H\odot\mathcal H$. The results hold for many weighted Besov spaces $\mathcal H$ in the unit ball of $\mathbb C^d$ provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that for weighted Besov spaces $\mathcal H$ that satisfy what we call the multiplier inclusion condition any bounded column multiplication operator $\mathcal H \to \oplus_{n=1}^\infty \mathcal H$ induces a bounded row multiplication operator $\oplus_{n=1}^\infty \mathcal H \to \mathcal H$. For the Drury-Arveson space $H^2_d$ this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions., Comment: minor changes

We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum modulus of the Taylor coefficients of a holomorphic function that maps the disk to the punctured disk is true.

We prove a factorization theorem for reproducing kernel Hilbert spaces whose kernel has a normalized complete Nevanlinna-Pick factor. This result relates the functions in the original space to pointwise multipliers determined by the Nevanlinna-Pick kernel and has a number of interesting applications. For example, for a large class of spaces including Dirichlet and Drury-Arveson spaces, we construct for every function $f$ in the space a pluriharmonic majorant of $|f|^2$ with the property that whenever the majorant is bounded, the corresponding function $f$ is a pointwise multiplier., Comment: 35 pages; minor changes

We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.

We prove a Montel theorem for Hilbert space valued functions, and a non-commutative version of this theorem, by composing with unitaries to achieve convergence.

This paper generalizes the classical Sz.-Nagy--Foias $H^{\infty}(\mathbb{D})$ functional calculus for Hilbert space contractions. In particular, we replace the single contraction $T$ with a tuple $T=(T_1, \dots, T_d)$ of commuting bounded operators on a Hilbert space and replace $H^{\infty}(\mathbb{D})$ with a large class of multiplier algebras of Hilbert function spaces on the unit ball in $\mathbb C^d$., Comment: 17 pages; minor changes

We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptano\u{g}lu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails., Comment: 18 pages; minor changes

We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This generalizes results of Carleson for the Hardy space and of Bishop, Marshall and Sundberg for the Dirichlet space. Furthermore, we investigate interpolating sequences for pairs of Hilbert function spaces.

We analyze the three point Pick interpolation problem on the bidisk

We give a characterization of interpolating sequences for bounded analytic functions on the bidisk.

We give a new treatment of Quiggin's and McCullough's characterization of complete Nevanlinna-Pick kernels. We show that a kernel has the matrix-valued Nevanlinna-Pick property if and only if it has the vector-valued Nevanlinna-Pick property. We give a representation of all complete Nevanlinna-Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.

The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carath\'eodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.

We describe those reproducing kernel Hilbert spaces of holomorphic functions on domains in ${\Bbb C}^d$ for which an analogue of the Nevanlinna-Pick theorem holds, in other words when the existence of a (possibly matrix-valued) function in the unit ball of the multiplier algebra with specified values on a finite set of points is equivalent to the positvity of a related matrix. Our description is in terms of a certain localization property of the kernel., Comment: in Proceedings of 17th International Conference on Operator Theory at Timisoara, 1998, Theta Foundation, Bucharest

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in $\mathbb{C}^d$. We look at differentiability at a boundary point for functions that have a certain regularity there.

For more than a decade, the debate over public school reform has created friction between teachers unions, administrators, school boards, parents, policymakers, and other stakeholders in public education and has fueled disagreements over how to improve the quality of teaching and learning for children. While many factors make consensus elusive when it comes to school reform, a key obstacle to finding agreement around educational improvements and bringing such improvements to fruition is ideological policy divisions. Yet within some districts and schools, union leaders and school administrators have found an alternate path to reform--one that is based on building strong relationships that facilitate collaboration among educators and is focused on teaching quality and educational improvement for students. This report explores the impact of school-level, union-management, institutional partnerships on teacher collaboration and student performance. Moreover, it offers strong evidence for this alternative direction to the policy debate on public school reform by analyzing the role of union-management relations in educational quality. This latest study finds that: (1) Formal partnerships help improve student performance; (2) Partnerships lead to more extensive communication between teachers; (3) More extensive communication improves student performance; and (4) Partnership leads to more frequent and informal communication between union representatives and principals. This report targets school-level union-management partnerships as potential catalysts for professional collaboration in public schools. Furthermore, this research sheds light on the impact that school-level union-management partnerships and teacher collaboration can have on student performance.

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.

We characterize functions of $d$-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = \sum a_n n^{-s-\bar u}$, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be "the same", and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury-Arveson space $H^2_d$ in $d$ variables, where $d$ can be any number in $\{1,2,\ldots, \infty\}$, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of $H^2_d$. Thus, a family of multiplier algebras of Dirichlet series are exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic to $H^2_d$ and when its multiplier algebra is isometrically isomorphic to $Mult(H^2_d)$., Comment: 23 pages

We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.

We establish rigidity (or uniqueness) theorems for nc automorphisms which are natural extensions of clasical results of H.~Cartan and are improvements of recent results. We apply our results to nc-domains consisting of unit balls of rectangular matrices.