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Open Access Publications from the University of California

Scholarly Works (16 results)

  • Article
  • Peer Reviewed
UC Santa Barbara Previously Published Works (2021)
Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.
    Cover page: Spectral dissection of finite rank perturbations of normal operators
    • Article
    • Peer Reviewed
    UC Santa Barbara Previously Published Works (2019)
    In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation of the resulting resolvent functional model is proposed.
      Cover page: A field-theoretic operator model and Cowen–Douglas class
      • Thesis
      • Peer Reviewed
      UC Santa Barbara Electronic Theses and Dissertations (2020)

      In this work we study the problem of writing a Hermitian polynomial as a Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea of Putinar-Scheiderer to obtain necessary matrix positivity conditions for Hermitian polynomials to be Hermitian sums of squares modulo Hermitian ideals. We show that the conditions are sufficient for a class of examples making a connection to the operator-valued Riesz-Fejer theorem and block Toeplitz forms. The work fits into the larger themes of Hermitian versions of Hilbert's 17-th problem and characterizations of positivity.

        Cover page: Hermitian Sums of Squares Modulo Hermitian Ideals
        • Thesis
        • Peer Reviewed
        UC Santa Barbara Electronic Theses and Dissertations (2020)

        Conjecture 0.1 (Conjecture of Blagovest Sendov (1958))

        For a complex polynomial of degree two or more with all its roots contained within the closed unit disk, each root has a critical point within unit distance.

        We introduce a countable collection of conjectures – one for each degree – by transferring into languages of Real Algebra. For fixed degree, each conjecture is decidable. Thus, we consider decidable statements in Real Algebra. For each of these decidable statements, we seek for certificates. We provide variations of this theme for different contexts: Positivstellensatz, Nichtnegativstellensatz and real radical ideal membership. In our context, our positivity certificates (or membership certificates) provide proof when achieved. We also find plenty of novel numerical evidence substantiating our conjectures.

        Keywords: Math, Maths, Mathematic, Mathematics, Mathematical, Pure Mathematics, Theoretical Mathematics, Applied Mathematics, Algorithm, Algorithms, Roots, Critical Points, Complex Analysis, Complex Polynomials, Real Algebra, Real Polynomials, Real Closed Field, Real Radical Ideal, Real Radical Ideal Membership, Ideal, Ideal Membership, Groebner, Gr ̈obner, Groebner Fan, Gr ̈obner Fan, Certificate, Certificate of Real Radical Ideal Membership, Certificate of Ideal Membership, Buchberger’s algorithm, S-Procedure, Real Algebraic Geometry, Algebraic Geometry, Linear Matrix Inequality, Linear Matrix Inequalities, LMI, Cylindrical Algebraic Decomposition, Procedure, Resultant, Lifting, Elimination, Input, Output, Semidefinite, Semidefinite Set, Semidefinite Sets, Semidefinite Programming, Nonlinear Programming, SDP, Relaxation Order, Semidefinite Relaxation, Slack Variable, Linear Operator, Linear Algebra, Moment, Moments, Moment Theory, Optimization, Optimum, Maximum, Supremum, Infimum, Minimum, Optimal, Maximal, Minimal, Extremal Polynomials, Convex Optimization, Convex Analysis, Functional Analysis, Function, Functional, Measure, Positivity, Positivity Certificate, Certificate, Polynomial, Positive Polynomial, Positive Polynomials, Sums of Squares, Ring, Preordering, Quadratic Module, Anneaux preordonnes, Rational Sums of Squares, SOS, Hilbert, David Hilbert, Artin, Emil Artin, Nullstellensatz, Positivestellensatz, Nichtnegativstellensatz, Nonnegative, Non-negative, Nonnegative Polynomial, Nonnegative Polynomials, Non-negative Polynomials, Tarski, Transfer Principle, Tarski-Transfer, Tarski-Seidenberg, Tarski-Seidenberg Principle, Dynamics, Nonlinear Dynamics, Fundamental Theorem of Algebra, Cauchy Integral Formula, The Argument Principle, Liouville’s Theorem, Rouche’s Theorem, Asymptotic, Asymptotics, Derivative, Differential Operator, Pullback, Complexity, Double Exponential, Computational Complexity, Numerics, Numerical, Decision, Decision Theory, Decision Problem, Decision Problems, Decidable, Axiom, Axiomatic, Independence, Model Theory, Higher-Order Logic,

        Conjecture, Blagovest Sendov, Sendov’s Conjecture, Ilief’s Conjecture, Duality, Duality Gap, Moment and Positive Polynomial Duality, Machine Error, Machine Precision, Validation, True, Truth, Incomplete, Incompleteness, Complete, Completeness, Contrapositive, Consistent, Formal Language, Language, Bounds, Global Optimum, Global Optimization, Constrained Optimization

        • 1 supplemental file
        Cover page: Roots and Critical Points of Complex Polynomials: Applications of Algorithms in Real Algebra, Moment Theory, Convex Analysis, Optimization, and Positive Polynomials to a Conjecture in Pure Mathematics
        • Article
        • Peer Reviewed
        UC Santa Barbara Previously Published Works (2019)
        Multivalued linear operators, also known as linear relations, are studied on a specific class of weighted, composition transforms on Fock space. Basic properties of this class of linear relations, such as closed graph, boundedness, complex symmetry, real symmetry, or isometry are characterized in simple algebraic terms, involving their symbols.
          Cover page: Multi-valued weighted composition operators on Fock space
          • Article
          • Peer Reviewed
          UC Santa Barbara Previously Published Works (2021)
          AbstractThe power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on which the power moment problem is accessible and solvable by non-negativity certificates.
            Cover page: The moment problem on curves with bumps
            • Article
            • Peer Reviewed
            UC Santa Barbara Previously Published Works (2020)
            We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a $\mathcal{C}$-selfadjoint operator generates a contraction $C_0$-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of \emph{complex symmetric, unbounded semigroups} and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all $\mathcal{C}$-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a $\mathcal{C}$-selfadjoint, unbounded semigroup is not necessarily $\mathcal{C}$-selfadjoint.
              Cover page: Complex symmetric evolution equations
              • Article
              • Peer Reviewed
              UC Santa Barbara Previously Published Works (2020)
              In two phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and to simplify the analysis Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored.
                Cover page: An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry
                • Article
                • Peer Reviewed
                UC Santa Barbara Previously Published Works (2021)
                Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.
                  Cover page: Data Analysis from Empirical Moments and the Christoffel Function
                  • Article
                  • Peer Reviewed
                  UC Santa Barbara Previously Published Works (2020)
                  Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel-Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an $N\times N$ Hermitian positive-definite Toeplitz matrix, where $N$ is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite-dimensional approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.
                    Cover page: Data-driven spectral analysis of the Koopman operator
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