Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

Name: Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations | 21st International Workshop on Operator Theory and Applications, Berlin, July 2010
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21st International Workshop on Operator Theory and Applications, Berlin, July 2010

Wolfgang Arendt Joseph A. Ball Jussi Behrndt Karl-Heinz Förster Volker Mehrmann Carsten Trunk(Editor)

Birkhäuser (Publisher)

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Published on 15. June 2012

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Intro
Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations
21st International Workshop on Operator Theoryand Applications, Berlin, July 2010
Contents
Preface
About the Gradient of the Conformal Radius
1. Introduction
2. Auxiliary results
3. Main results
References
The Influence of the Tunnel Effect on L8-time Decay
1. Introduction
2. A solution formula
3. L8-time decay
4. Growing potential step
Acknowledgement
References
Some Classes of Operators on Partial Inner Product Spaces
1. Introduction
2. Partial inner product spaces
2.1. Basic definitions
2.2. Examples
2.2.1. Sequence spaces
2.2.2. Lebesgue spaces
2.2.3. Spaces of locally integrable functions
3. Operators in pip-spaces
3.1. Definitions
3.2. Regular operators
3.3. Morphisms
3.3.1. Monomorphisms
3.3.2. Epimorphisms
3.3.3. Isomorphisms
3.3.4. Orthogonal projections.
4. Examples of operators
4.1. General operators
4.1.1. Regular linear functionals
4.1.2. Dyadics
4.1.3. Matrix elements
4.2. Integral operators on Lp spaces
5. LHS generated by O*-algebras
6. Conclusion
References
From Forms to Semigroups
Introduction
1. The Hille-Yosida Theorem
2. Holomorphic semigroups
3. The Lumer-Phillips Theorem
4. Forms: the complete case
5. The Stokes operator
6. From forms to semigroups: the incomplete case
7. Degenerate diffusion
8. The Dirichlet-to-Neumann operator
References
Accretive (*)-extensions and Realization Problems
1. Introduction
2. Preliminaries
3. The Friedrichs and Kre.in-von Neumann extensions
4. Bi-extensions of non-negative symmetric operators
5. Accretive bi-extensions
6. Realization of Stieltjes functions
7. Realization of inverse Stieltjes functions
8. Examples
Acknowledgment
References
The Dynamical Problem for a Non Self-adjoint Hamiltonian
1. Introduction
2. Quantization recipes using non-unitary Dyson mappings O
2.1. The three-space scenario
2.2. A redefinition of the conjugation
2.3. PT-symmetric quantum mechanics
3. Summary
Acknowledgment
References
Extension of the ?-metric: the H8 Case
1. Introduction
2. Recap of the abstract ?-metric
2.1. The notation F(R)
2.2. The notation F*
2.3. Right coprime/normalized coprime factorization
2.4. Left coprime/normalized coprime factorization
2.5. The notation G,G,K,K
2.6. The notation S(R, p,m)
3. The ?-metric when R = QA
References
Families of Homomorphisms in Non-commutative Gelfand Theory: Comparisons and Examples
1. Introduction and preliminaries
2. First observations
3. Matrix representations
4. Separating versus sufficient families of matrix representations
5. Separating versus weakly sufficient families of matrix representations
Acknowledgment
References
Operator Splitting with Spatial-temporal Discretization
1. Introduction
2. Rational approximations
Proof of Lemma 2.1
Acknowledgment
References
On a Theorem of Karhunen and Related Moment Problems and Quadrature Formulae
1. Introduction
2. Karhunen's theorem
3. A necessary and sufficient condition
4. The second problem
4.1. The trigonometric moment problem
4.2. The power moment problem
4.3. A non-elementary counterexample
4.4. An elementary counterexample
5. Molecular measures and a quadrature formula
Acknowledgment
References
Explicit Error Estimates for Eigenvalues of Some Unbounded Jacobi Matrices
1. Introduction
Assumptions
Estimating the nth eigenvalue
2. Main results
3. Finite Jacobi matrices
3.1. Notations
3.2. Application of the min-max principle
3.3. Estimate of the middle eigenvalue
4. Middle eigenvalue estimate: a first improvement
4.1. Main result of this section
4.2. Solution of a commutator equation
4.3. Auxiliary integral representations
4.4. Proof of Proposition 4.1
5. Middle eigenvalue estimate: a second improvement
5.1. Main result of this section
5.2. The operator J'1= J1 + R
5.2.1. We introduce a first correction term
5.2.2. We introduce a second correction term
5.2.3. Let us now define the correction
5.3. Analysis of R
5.4. Proof of Proposition 5.1
6. Successive approximations
6.1. Main result of this section
6.2. First step (k = 1)
6.3. Induction step (k = 2, . . . ,m)
6.3.1. Construction of Dk
6.3.2. Construction of Ak
6.3.3. Construction of Rk
6.4. Estimate of R2
6.5. Proof of Proposition 6.1 (end)
6.6. Middle eigenvalue estimate: last improvement
6.7. Computing an approximation of the middle eigenvalue
7. Proofs of the main results
7.1. Notations
7.2. Proof of Theorem 2.1
7.3. Estimates by middle eigenvalues of finite Jacobi submatrices
7.4. Proofs of Theorems 2.2 and 2.3
7.5. Proof of Theorem 2.4
References
Algebraic Reflexivity and Local Linear Dependence: Generic Aspects
1. Introduction
2. Overview and results
2.1. Preliminaries
2.2. Reflexivity
2.3. Local linear dependence
2.4. Generic reflexivity
2.5. The basic relation
2.6. Total reflexivity
3. Equality in the inclusions S?Sgref ? Mn,m
4. Proofs of Theorems 2.7 and 2.13
5. Properties of localized span
6. Canonical forms for a matrix pencil
7. Genericity-related classification of divisors
8. Proof of Theorems 2.3, 2.10 and 2.14
9. Can the product AB belong to [A,B]ref \ [A,B]?
Acknowledgment
References
An Invitation to the S-functional Calculus
1. Introduction
2. Slice monogenic functions
3. The functional setting and preliminary results
4. The S-functional calculus for bounded operators
5. The SC-functional calculus
5.1. The F-spectrum is easier to compute with respect to the S-spectrum
6. The F-functional calculus for bounded operators
7. The S-functional calculus for unbounded operators
References
Necessity of Parameter-ellipticity for Multi-order Systems of Differential Equations
1. Introduction and main results
2. Proof of the necessity
References
Canonical Eigenvalue Distribution of Multilevel Block Toeplitz Sequences with Non-Hermitian Symbols
1. Introduction and basic notations
2. Preliminary definitions and tools
3. Further tools for general matrix sequences and main results
4. Proof of the main result for the case of block Toeplitz sequences and their algebra
5. Numerical experiments
5.1. Examples covered by the theory
5.2. Examples with unbounded symbols
6. Concluding remarks and open problems
References
Curvature Invariant and Generalized Canonical Operator Models - I
1. Introduction
2. Hilbert modules
3. The class Bn(D)
4. Proof of the main results
5. Concluding remark
References
About Compressible Viscous Fluid Flow in a 2-dimensional Exterior Domain
1. Introduction
2. An analytic semigroup approach to the compressible viscous fluid flow
3. Some decay properties of the Stokes semigroup in a 2-dimensional exterior domain
4. Global in time unique existence theorem of strong solution to (1.1) in a 2-dimensional exterior domain
Acknowledgment
References
On Canonical Solutions of a Moment Problem for Rational Matrix-valued Functions
0. Introduction
1. On canonical nonnegative definite sequences of matrices and corresponding matrix measures
2. On canonical solutions of Problem (R), the general case
3. Descriptions of canonical solutions in the nondegenerate case
4. Some conclusions from Theorem 3.6
Some final remarks
References
Maximal Lp-regularity for a 2D Fluid-Solid Interaction Problem
1. Introduction
2. Model and main result
3. Preliminary results
4. A reformulation of the problem
5. Proof of Theorem 1
References
Transfer Functions for Pairs of Wandering Subspaces
Introduction
1. Pairs of subspaces
2. Examples: Characteristic functions
2.1. Example
2.2. Example
3. Examples: Noncommutative Markov chains
Acknowledgment
References
Non-negativity Analysis for Exponential-Polynomial Trigonomet Functions on [0, 8)
1. Introduction
2. Non-negativity analysis on a finite interval - a summary
3. A special representation of EPT functions
4. A projection on the boundary of a set of non-negative functions
5. Conclusions and further research
Acknowledgement.
References
Compactness of the ?-Neumann Operator on Weighted (0,q)-forms
1. Introduction
2. The weighted Kohn-Morrey formula
3. Compactness of Nf,q
References
Dislocation Problems for Periodic Schrödinger Operators and Mathematical Aspects of Small Angle Grain Boundaries
1. Introduction
2. Dislocation problems on the real line, on the strip R × [0, 1], and in the plane
2.1. Lemma
2.2. Proposition
2.3. Theorem
2.4. Theorem
2.5. Theorem
2.6. Theorem
3. Rotational defect in a two-dimensional lattice
3.1. Lemma
4. Muffin tin potentials
4.1. Proposition
4.2. Remark
4.3. Proposition
Acknowledgement
References
The Riemann Boundary Value Problem on Non-rectifiable Arcs and the Cauchy Transform
Introduction
1. Integrations and dimensions
2. The Cauchy transform of integrations
3. The Riemann boundary value problem on non-rectifiable arcs
References
Decay Estimates for Fourier Transforms of Densities Defined on Surfaces with Biplanar Singularities
1. Decay of Fourier transforms
2. Applications to crystal theory
References
Schatten-von Neumann Estimates for Resolvent Differences of Robin Laplacians on a Half-space
1. Introduction
2. Quasi-boundary triples and Robin Laplacians on a half-space
3. Compactness and Schatten-von Neumann estimates for resolvent differences of Robin Laplacians
References
Smoothness of Hill's Potential and Lengths of Spectral Gaps
1. Introduction
2. Main result
3. Preliminaries
4. The proofs
4.1. Proof of Theorem 1
Appendix A. Hörmander spaces on the circle
References
A Frucht Theorem for Quantum Graphs
1. Introduction
2. Quantum graphs
3. Symmetries of quantum graphs
4. Isospectral non-isomorphic graphs
References
Note on Characterizations of the Harmonic Bergman Space
1. Introduction
2. Preliminaries
3. Proof of the main theorem
References
On Some Boundary Value Problems for the Helmholtz Equation in a Cone of 240°
1. Introduction
2. Reduction to a pair of problems in convex cones (half-angle)
3. Sommerfeld potentials
4. Solution of the reduced semi-homogeneous BVPs
5. Solution of the BVPs with symmetry in O = Oo,a
6. Analytic solution of DD and NN problems in O = Oo,a
Acknowledgment
References
The Infinite-dimensional Sylvester Differential Equation and Periodic Output Regulation
1. Introduction
2. Notation and definitions
3. The infinite-dimensional Sylvester differential equation
4. Periodic Output Regulation
5. Conclusions
References
A Class of Evolutionary Problems with an Application to Acoustic Waves with Impedance Type Boundary Conditions
0. Introduction
1. General solution theory
2. Causality
3. An application: Acoustic waves with impedance type boundary condition
4. Conclusion
References
Maximal Semidefinite Invariant Subspaces for J-dissipative Operators
1. Introduction
2. Preliminaries
3. Main results
References
The Riemann-Hilbert Boundary Value Problem with a Countable Set of Coefficient Discontinuities and Two-side Curling at Infinity
1. Introduction
2. Statement of the problem
3. Solvability of the homogeneous Riemann-Hilbert boundary value problem
4. The inhomogeneous problem
References
Galerkin Method with Graded Meshes for Wiener-Hopf Operators with PC Symbols in Lp Spaces
1. Introduction
2. Spaces definitions
3. Projections and related operators
4. Approximation methods
5. Algebraization
6. Essentialization
7. Localization
8. Identification of the local algebras
References
On the Spectrum of Some Class of Jacobi Operators in a Krein Space
1. Introduction
2. Self-similar functions of zero spectral order
3. Sturm-Liouville problem with discrete self-similar weight
3.1. Discretization of eigenfunctions of Sturm-Liouville problem with discrete self-similar weight
3.2. Indefinite case
References
Holomorphy in Multicomplex Spaces
1. Introduction
2. The 2-dimensional bicomplex space
3. The nth multicomplex space
4. Holomorphy in BC3
5. Recursive formulas
6. Algebraic properties of the sheaf of holomorphic functions on BCn
Acknowledgment
References
On Some Class of Self-adjoint Boundary Value Problems with the Spectral Parameter in the Equations and the Boundary Conditions
1. Introduction
2. Formulation of the Abstract General Problem (AGP)
3. Properties of the abstract boundary value problems
4. Asymptotic formulas for the branch of positive eigenvalues
5. Multicomponent transmission spectral problems
Acknowledgment
References
On the Well-posedness of Evolutionary Equations on Infinite Graphs
1. Introduction
2. Evolutionary equations in mathematical physics
3. Systems with integration by parts
4. Applications
Acknowledgment
References
Reparametrizations of Non Trace-normed Hamiltonians
1. Introduction
2. Hamiltonians and their operator models
2.1. Definition
2.3. Definition
2.4. Lemma
2.5. Corollary
2.6. Definition
2.7. Definition
3. Absolutely continuous reparametrizations
3.1. Definition
3.4. Proposition
3.5. Lemma
3.6. Lemma
3.7. Proposition
3.8. Theorem
4. Trace-normed and non-vanishing Hamiltonians
a. Existence of trace-norming reparametrizations
4.1. Lemma
4.2. Proposition
4.4. Corollary
b. Description of '~' for non-vanishing Hamiltonians
4.5. Lemma
4.6. Lemma
4.7. Lemma
4.8. Lemma
4.9. Proposition
References

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