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Positive Operators

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

La reproductivite nette R0 est utilisee en biologie des populations et notamment en epidemiologie depuis plusieurs decennies. Mais on n'a propose une definition convenant au cas des modeles avec coefficients periodiques qu'il y a quelques annees. La definition fait intervenir le rayon spectral d'un operateur integral. Comme dans l'etude des modeles epidemiques structures dans un environnement constant, il est bon d'expliquer la signification biologique de ce rayon spectral. On montre dans cet article que R0 pour les modeles periodiques est encore un taux asymptotique de croissance par generation. On insiste aussi sur la difference entre ce R0 theorique pour les modeles periodiques et la ≪ reproductivite nette ≫ obtenue en ajustant une exponentielle au debut d'une courbe epidemique. Les etudes recentes sur la pandemie de grippe H1N1 n'ont pas pris en compte cette difference.

https://hal.archives-ouvertes.fr/hal-01274025/document

Genealogy with seasonality, the basic reproduction number, and the influenza pandemic.

We show that every $$n\,\times \,n$$n×n matrix is generically a product of $$\lfloor n/2 \rfloor + 1$$?n/2?+1 Toeplitz matrices and always a product of at most $$2n+5$$2n+5 Toeplitz matrices. The same result holds true if the word `Toeplitz' is replaced by `Hankel,' and the generic bound $$\lfloor n/2 \rfloor + 1$$?n/2?+1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary $$(2n-1)$$(2n-1)-dimensional subspace of $${n\,\times \,n}$$n×n matrices. Furthermore, such decompositions do not exist if we require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.

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Every Matrix is a Product of Toeplitz Matrices

Let (X, μ) be a σ-finite measure space and M(X × X)+ a cone of all equivalence classes of (almost everywhere equal) non-negative measurable functions on a product measure space X × X. Using Luxemburg-Gribanov theorem we define a product * on M(X × X)+. Given a function seminorm h : M(X × X)+ → [0, ∞], we introduce the spectral radius rh (f) of f ∈ M(X × X)+ with respect to h and *. We give several examples. In particular, rh (f) provides a generalization and unification of the spectral radius and its max version, the maximum cycle geometric mean, of a non-negative matrix.

Inequalities for the spectral radius of non-negative functions

Let$\mathcal{C}$ be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that$\mathcal{C}$ is finitely quasinilpotent at a vectorx0∈X whenever for any finite subset$\mathcal{F}$ of$\mathcal{C}$ the joint spectral radius of$\mathcal{F}$ atx0 is equal 0. If such collection$\mathcal{C}$ contains a non-zero compact operator, then$\mathcal{C}$ and its commutant$\mathcal{C}'$ have a common non-trivial invariant, subspace. If in addition,$\mathcal{C}$ is a collection of positive operators on a Banach lattice, then$\mathcal{C}$ has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let$\mathcal{S}$ be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then$\mathcal{S}$ has a common non-trivial invariant closed ideal.

Common invariant subspaces for collections of operators

Abstract In the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space (of dimension greater than 1) possessing an atom. For instance, if S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invariant closed ideal. Furthermore, if T is a non-zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ≤ ST for all S ∈ S, then S and T have a common non-trivial invariant closed ideal. We also give a simple example of a quasinilpotent compact positive operator on the Banach lattice l∞ with no non-trivial invariant band. The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L. For a semigroup S of quasinilpotent, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, then this chain is maximal in the lattice of all closed subspaces of L.

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Triangularizing semigroups of positive operators on an atomic normed Riesz Space

Let K1, . . . , Kn be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K1, . . . , Kn, the operator K, satisfies the following inequalities

Inequalities for the hadamard weighted geometric mean of positive kernel operators on banach function spaces

K. M. R. Audenaert (2010), R. A. Horn and F. Zhang (2010), Z. Huang (2011), A. R. Schep (2011), A. Peperko (2012), and D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite nonnegative matrices that define operators on sequence spaces. In the present article, we extend and refine several of these results, and we also prove some analogues for the numerical radius.

Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

Let A and B be positive operators on a Banach lattice such that the commutator C = AB − BA is also positive. We study the size of the spectrum of C.

Roman Drnovšek

Papers

Common invariant subspaces for collections of operators

Triangularizing semigroups of positive operators on an atomic normed Riesz Space

Inequalities for the hadamard weighted geometric mean of positive kernel operators on banach function spaces

Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

On positive commutators