Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis: IWOTA 2021, Chapman University

Preface
Contents
Infinite Order Differential Operators with a Glimpse to Applications to Superoscillations
1 Introduction: Infinite Order Differential Operators and Convolution Operators
2 Spaces of Entire Functions
3 Infinite Order Differential Operators and Superoscillations
4 Hypercomplex Case: Cauchy-Fueter and Monogenic Case
References
Interpolation in Multivariable de Branges-Rovnyak Spaces
1 Introduction
2 The Classical Vector-Valued Case
2.1 OAPH(S0) as a Generalization of the Beurling-Lax Theorem
3 The Fock-Space Setting
3.1 Interpolation Problem
3.2 Noncommutative Formal Reproducing Kernel Hilbert Spaces
3.3 Indefinite Noncommutative Schur Class
3.4 Linear-Fractional Maps Associated with Indefinite Noncommutative Schur-Class Multipliers
3.4.1 Injectivity (or Lack Thereof) of Linear-Fractional Maps
3.4.2 Construction of Indefinite Schur-Class Multipliers
3.5 OAPH(S0): Parametrization of the Solution Set
3.6 Interpolation in H2Y(F+d)
4 The Drury-Arveson Space Setting
5 An Operator Theoretical View
5.1 Generalized de Branges-Rovnyak Spaces
5.2 The Solutions to Douglas Factorization Problem
5.3 Douglas Factorization Problem in Generalized de Branges-Rovnyak Spaces
Appendix 1: A Kreĭn Space Lemma
Appendix 2: Schur Multipliers and Their Adjoints
References
Monotonicity of Certain Left and Right Riemann Sums
1 Introduction
2 Background and Context
3 The Left Riemann Sum of 11+x2
4 A Finer Analysis via Experimentation
5 Riemann Sum of Trigonometric Functions
6 Some Prerequisites
7 Proof of Theorem 12
8 Concluding Remarks
References
A Survey on the Recent Advances in the Spectral Theory on the S-Spectrum
1 Introduction
2 Slice Hyperholomorphic Functions and the Flexibility of Their Cauchy Formulas
2.1 Spaces of Hyperholomorphic Functions
2.2 The Flexibility of the Cauchy Formula of Slice Hyperholomorphic Functions
3 Slice Hyperholomorphic Functions of a Clifford Variable
4 Vector-Valued Slice Hyperholomorphic Functions
5 The Abstract Formulation of the S-Functional Calculus
5.1 Application: Noncommuting Matrix Variables
6 The Spectral Theorem for Clifford Operators
6.1 Basic Facts for Normal Operators
6.2 Measure Theory and Integration Theory for Rn-Valued Measures
6.3 Spectral Integrals
6.4 Basic Properties of Spectral Integrals
6.5 Spectral Integrals of Bounded Measurable Functions
6.6 Spectral Integrals of Unbounded Measurable Functions
6.7 The Spectral Theorem for a Bounded Self-Adjoint Operator
6.8 Polar Decomposition for Bounded Operators
6.9 An Additive Decomposition for Bounded Operators and Imaginary Operators
6.10 The Spectral Theorem for a Bounded Normal Operator
7 Some Concluding Remarks
References
Stochastics and Dynamics of Fractals
1 Introduction
2 Analysis of Cantor Measures
2.1 Applications to Iterated Function System Measures
2.2 Gaussian Processes
2.3 IFS Measures Supported on Compact Intervals
2.4 Leibnitz' Rule and Functional Calculus
2.5 Skew Symmetric vs Skew-Adjoint
2.6 More About g-1=λ1
2.7 V maps L2(λ) onto L2()
3 Results for Reproducing Kernel Hilbert Spaces (RKHS) Associated to General Measure Spaces
3.1 The Generalized Wiener-Process
3.2 Gaussian Fields
3.2.1 The Path-Space Ω
3.2.2 L2()-factorizations
3.3 Krein-Feller Diffusion
3.4 Signed Measures: Finitely Additive vs Countably Sigma-Additive
4 Dual Pairs
References
Fluid–Plate Interaction with Kelvin-Voigt Damping and Bending Moment at the Interface: Well-posedness, Spectral Analysis, Uniform Stability
1 Introduction
2 The Coupled PDE Model. A Physical 2-Dimensional Visco-Elastic Plate Coupled with Fluid at the Interface
3 Abstract Model of Problem (2.1a–2.1g). The Free Dynamics Operator A
4 Main Results
5 Proof of Theorem 4.1: Dissipativity of A and A* and Generation
5.1 The Operator A
5.2 The Adjoint Operator A*: Definition and Dissipativity
6 Proof of Theorem 4.2: Analyticity of eA t and eA* t
7 Proof of Theorem 4.3
8 The Subspace H"0362H= [ N(A)] =[ N(A*)] of Codimension 1; Its Invariance Under the Semigroups eA t or eA*t; Their Uniform Stability on H"0362H
Appendix 1: Spectral (Eigenvalue) Analysis
Appendix 2: A Physically Attractive Case Where the B.C B1 wt|s in (2.1f) Is Replaced by [ B1(w+wt)]|s
References
Automorphisms of Hyper-Reinhardt Free Spectrahedra
1 Introduction
1.1 Prologue
1.2 Free Spectrahedra
1.3 Free Maps
1.4 Hyper-Reinhardt Free Spectrahedra
1.5 Context and Prior Results
1.5.1 Spectraballs, Circular Symmetry and Prior Results
1.5.2 Complete Positivity
1.5.3 Automorphism Groups and Change of Variables
2 Carathéodory Interpolation Preliminaries
3 The Affine Linear Terms
4 The Higher Order Terms
4.1 The Hyper-Reinhardt Condition
4.2 Proof of Lemma 4.2
4.3 The Sets N and Z
4.4 Summary
5 Normalized Automorphisms and Compositions
6 The Case of the Identity Permutation
6.1 Two Auxiliary Reinhardt Spectrahedra
6.2 Two Auxiliary Automorphisms
6.3 How to Spot a Polydisc
7 Proof of Theorem 1.1
7.1 Proof of Item (i)
7.2 Proof of Item (iii)
References
Arithmetic and Analysis of the Series n=1∞ 1n sinxn, Part II
1 Introduction
2 The Origin of the Function of Flett, Hilbert Space Method
3 Euler, Bessel, Mellin and Beyond
3.1 A Remarkable Formula and the Cube of a Theta Function
4 Summations Formulas and Beyond
4.1 Some Classical Choices
4.2 Another Important Choice a(n)= r2(n)
4.3 Explicit Formula and Extension of (1.2)
5 Unified Approach, Eisenstein Series
5.1 Specializations
6 von Mangoldt Twist n=1∞ (n)n(e-zn-1) and Riemann-von Mangold Function
7 Lattice Sums, r3(n) and Madelung's Constant
7.1 Cubic Lattice Sums
8 The Forms ψ(x,y,z)= x2+y2+z2 and φ(x,y,z)= xy+yz+zx, Lacunarity
8.1 The Two Forms
8.2 Sum of Squares
8.3 Linnik Problem
9 An Interesting Link and a Remark
References
Lipschitz-Type Bounds for Functions of Operators with Noncompact Perturbations
1 Introduction
2 Lipschitzness from Sp to Sp, p≥1
3 Modified Lipschitzness from B(H) to Sp, p≥1
3.1 General Operator Theoretic Setup
3.2 Applications to Schrödinger and Dirac Operators
4 Remarks on Methods
References
Extended Fock Space Formalism and Polyanalytic Functions
1 Introduction
2 Extended Fock Space Formalism
2.1 Generic Case
2.2 Special Case of b = a†
2.3 On the Operator aa†
3 Unit Disk D
3.1 First Approach
3.2 Second Approach
4 Complex Plane C
5 Some (Counter)Examples
References
Estimates of Cauchy–Szegö Kernel in Hardy Spaces on Nilpotent Lie Groups of Step Two
1 Atomic Decomposition of Hardy Spaces on Nilpotent Lie Groups of Order Two
2 Applications to the Hp Boundedness of NIS Operators
3 Examples of NIS Operators on Nilpotent Lie Groups of Step Two
3.1 Laguerre Calculus on Nilpotent Lie Groups of Step Two
3.2 Twisted Convolution
3.3 The Laguerre Basis
3.4 The Fundamental Solution to the Sub-Laplacian
3.5 The Szegö Kernel on the Quaternionic Heisenberg Group
References