Schur Class Research Articles

Hodge-Riemann bilinear relations for Schur classes of ample vector bundles

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class $s_{\lambda}(E)\in H^{d-2,d-2}(X)$ has the Hard Lefschetz property and satisfies the Hodge-Riemann bilinear relations. As a consequence we obtain various new inequalities between characteristic classes of ample vector bundles, including a higher-rank version of the Khovanskii-Teissier inequalities.

Hodge–Riemann Relations for Schur Classes in the Linear and Kähler Cases

Abstract We prove a version of the Hodge–Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.

Nonuniqueness of Carathéodory Extremal Functions on the Symmetrized Bidisc

We survey the Carathéodory extremal problem Car δ on the symmetrized bidisc $$G\,\,\mathop = \limits^{{\rm{def}}} \,\,\left\{ {\left( {z + w,zw} \right):\left| z \right| < 1,\,\,\,\left| w \right| < 1} \right\} = \left\{ {\left( {s,p} \right) \in {\mathbb{C}^2}:\left| {s - \bar sp} \right| < 1 - {{\left| p \right|}^2}} \right\}.$$We also give some new results on this topic. We are particularly interested in cases of this problem in which the solution of the problem is not unique. It is known that, for any δ = (λ, v) ∈ TG with v ≠ 0, there is at least one \(\omega \in \mathbb{T}\) such that Φω solves Car δ, where \({\Phi _\omega }\left( {s,p} \right) = {{2\omega p - s} \over {2 - \omega s}}\). Moreover, there is an essentially unique solution of Car δ if and only if δ has exactly one Carathéodory extremal function of the form Φω for some \(\omega \in \mathbb{T}\). We give a description of Carathéodory extremals for δ ∈ TG with more than one Carathéodory extremal function Φω for some values of \(\omega \in \mathbb{T}\). The proof exploits a model formula for the Schur class of G which is an ana log of the well-known network realization formula for Schur-class functions on the disc.

Discrete analytic Schur functions

We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem.

Analytical continuation of matrix-valued functions: Carathéodory formalism

Finite-temperature quantum field theories are formulated in terms of Green's functions and self-energies on the Matsubara axis. In multi-orbital systems, these quantities are related to positive semidefinite matrix-valued functions of the Carath\'eodory and Schur class. Analysis, interpretation and evaluation of derived quantities such as real-frequency response functions requires analytic continuation of the off-diagonal elements to the real axis. We derive the criteria under which such functions exist for given Matsubara data and present an interpolation algorithm that intrinsically respects their mathematical properties. For small systems with precise Matsubara data, we find that the continuation exactly recovers all off-diagonal and diagonal elements. In real-materials systems, we show that the precision of the continuation is sufficient for the analytic continuation to commute with the Dyson equation, and we show that the commonly used truncation of off-diagonal self-energy elements leads to considerable approximation artifacts. Our method paves the way for the systematic evaluation of Matsubara data with equations of many-body theory on the real-frequency axis.

On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n, and let E be an ample vector bundle over X. We show that any Schur class of E, lying in the cohomology group of bidegree (n − 1, n − 1), has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld (1983) for certain degrees.

Exponentials of de Branges–Rovnyak kernels

AbstractIn this note, we give a new property of de Branges–Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges–Rovnyak kernel is strictly positive definite if the corresponding Schur class function is nontrivial.

Bergman spaces over noncommutative domains and commutant lifting

The goal of the present paper is to provide analogues of Sarason interpolation theorem in the Hardy algebra H∞(D) and Sz.Nagy-Foiaş commutant lifting theorem for contractions on Hilbert spaces in the setting of noncommutative Hardy spaces associated with noncommutative regular domains and varieties. This is accompanied by the study of multi-analytic operators with respect to the universal models associated with the regular domains (resp. varieties) and the study of multipliers of noncommutative Bergman spaces.As applications, we obtain Toeplitz-corona theorems for multi-analytic operators, commutant lifting in several variables where the liftings are in certain Schur classes, factorization of multi-analytic operators, and Nevanlinna-Pick interpolation results for multipliers of Bergman spaces over Reinhardt domains in Cn. In addition, we obtain Andô type dilations and inequalities on noncommutative bi-domains and varieties.

Some structural and closure properties of an extension of the q-tensor product of groups,

In this work we study some structural properties of the group q a non-negative integer, which is an extension of the q-tensor product where G and H are normal subgroups of some group L. We establish by simple arguments some closure properties of when G and H belong to certain Schur classes. This extends similar results concerning the case q = 0 found in the literature. Restricting our considerations to the case G = H, we compute the q-tensor square for q odd, where Dn denotes the dihedral group of order 2n. Upper bounds for the exponent of are also established for nilpotent groups G of class which extend to all similar bounds found by Moravec, P. (2008). The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra. 212(7):1840–1848.

Compressed Resolvents, Schur Functions, Nevanlinna Families and Their Transformations

We study certain transformations of Nevanlinna families and Schur class operator-valued functions, construct their realizations by means of compressed resolvents and by the transfer functions of conservative systems, and find fixed points of such transformations. In particular, we characterize some automorphic Schur functions and corresponding them Nevanlinna families, including periodic and scale invariant.

On the Nevanlinna problem: characterization of all Schur–Agler class solutions affiliated with a given kernel

Given a domain $\Omega$ in $\mathbb{C}^m$, and a finite set of points $z_1,z_2,\ldots, z_n\in \Omega$ and $w_1,w_2,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the \textit{Pick interpolation problem} asks when there is a holomorphic function $f:\Omega \rightarrow \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ to exist if $\Omega=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that \textit{interpolate} the data. We generalize Nevanlinna's result to a domain $\Omega$ in $\mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.

Characterization of the Schur class in terms of the coefficients of a series on the Laguerre basis

The classical Schur criterion answers the question of whether a function f given by its power series f(x)=∑k=0∞CkZk is a Schur function that is, holomorphic in a unit disk D and such that supz∈D | f (z) | ≤ 1. Regarding this criterion, there are a large number of completed results devoted to its generalizations and various applications, but, as it seems to us, there is no criterion for a complete description of the Schur class in terms of coefficients of orthogonal series on arbitrary complete orthonormal systems. In this paper, we formulate such criterion for a formal orthogonal series with complex coefficients based on the Laguerre system.

Schur Parameters and the Carathéodory Class

The Schur (resp. Caratheodory) class consists of all the analytic functions f on the unit disk with $$|f|\le 1$$ (resp. $${\,{\text {Re}}\,}f>0$$ and $$f(0)=1$$). The Schur parameters $$\gamma _0,\gamma _1,\dots (|\gamma _j|\le 1)$$ are known to parameterize the coefficients of functions in the Schur class. By employing a recursive formula for it, we describe the n-th coefficient of a Caratheodory function in terms of n independent variables $$\gamma _1,\dots , \gamma _n$$. The mapping properties of those correspondences are also studied.

Aleksandrov–Clark Theory for Drury–Arveson Space

Recent work has demonstrated that Clark’s theory of unitary perturbations of the backward shift on a deBranges–Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury–Arveson multiplier algebra and the Aleksandrov–Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Free Disk operator system $$\mathcal {A} _d + \mathcal {A} _d ^*$$ , where $$\mathcal {A} _d$$ is the Free or Non-commutative Disk Algebra on d generators. We continue this program for vector-valued Drury–Arveson space by establishing the existence of a canonical ‘tight’ extension of any Aleksandrov–Clark map to the full Free Disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov–Clark maps.

Non-commutative Clark measures for the free and Abelian Toeplitz algebras

We construct a non-commutative Aleksandrov–Clark measure for any element in the operator-valued free Schur class, the closed unit ball of the free Toeplitz algebra of vector-valued full Fock space over Cd. Here, the free (analytic) Toeplitz algebra is the unital weak operator topology (WOT)-closed algebra generated by the component operators of the free shift, the row isometry of left creation operators. This defines a bijection between the free operator-valued Schur class and completely positive maps (non-commutative AC measures) on the operator system of the free disk algebra, the norm-closed algebra generated by the free shift. Identifying Drury–Arveson space with symmetric Fock space, we determine the relationship between the non-commutative AC measures for elements of the operator-valued commutative Schur class (the closed unit ball of the WOT-closed Toeplitz algebra generated by the Arveson shift) and the AC measures of their free liftings to the free Schur class.

Harnack and Shmul’yan pre-order relations for Hilbert space contractions

We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul’yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul’yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul’yan parts coincide. We also discuss an extension, recently considered by S. ter Horst (2014), of the Shmul’yan pre-order from contractions to the operator-valued Schur class of functions. In particular, the Shmul’yan–ter Horst part of a given partial isometry, viewed as a constant Schur class function, is explicitly determined.

On the Relationships Between Central Series in Some Locally Finite Groups

It is shown that the class of locally finite divisible-by-bounded groups is both a Schur class and a Baer class.

Analytic functions on the bidisk at boundary singularities via Hilbert space methods

We investigate the behavior of a generalized Hilbert space model of a function in the Schur class of the bidisk at singular boundary points that satisfy a growth condition. We examine the relationship between the boundary behavior of Schur functions and the geometry of corresponding generalized Hilbert space models. We describe a geometric condition on an associated operator that classifies the behavior of the directional derivative of the underlying Schur function at a carapoint.

A rich structure related to the construction of analytic matrix functions

We study certain interpolation problems for analytic 2×2 matrix-valued functions on the unit disc. We obtain a new solvability criterion for one such problem, a special case of the μ-synthesis problem from robust control theory. For certain domains X in C2 and C3 we describe a rich structure of interconnections between four objects: the set of analytic functions from the disc into X, the 2×2 matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. This rich structure combines with the classical realisation formula and Hilbert space models in the sense of Agler to give an effective method for the construction of the required interpolating functions.

A generalized tangent interpolation problem

A new tangent interpolation problem in the Schur class is studied. For this problem, the necessary and sufficient conditions of its solvability are obtained, and the description of solutions is given. The results are applied to the classification and the description of the set of exceptional parameters of an indefinite tangent interpolation problem in generalized Schur classes.