关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed "projected consensus algorithm" in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed "projected subgradient algorithm" which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set.

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Constrained Consensus and Optimization in Multi-Agent Networks

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Positive Definite Matrices

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

A hilbert space problem book

Preface .- I. Banach Algebras .- II. Operators .- III. Essential Spectrum .- IV. Taylor Spectrum .- V. Orbits and Capacity .- Appendix .- Bibliography

Spectral Theory of Linear Operators: And Spectral Systems in Banach Algebras

An analogue to the Sz.-Nagy-Foia5 dilation theory is presented for several commuting operators on a Hilbert space. The dilation theory of Hilbert space contractions, a reference of which is the excellent monograph [9], has not yet found a complete counterpart valid for several (commuting) Hilbert space operators. Apart from the results of this type already contained in [9] and other results of general character (i.e., Arveson's or Stinespring's extension theorem), there are not too many contributions specialized to the multiparameter dilation theory. Among these contributions, we quote the works [7, 3-5, 10], which are closer to our topics. The aim of this paper is to analyse some positivity conditions for commuting multioperators, which ensure the unitary equivalence of these objects to some standard models consisting of backwards multishifts. We emphasize, in particular, a situation that seems to satisfy the requirements of an appropriate extension for the dilation theory of Hilbert space contractions and that exploits the results already obtained in [4, 10]. Our finer methods also imply better statements of some assertions from [10] and provide an answer to Question 4.5 from [5]. Let n > 1 be an integer, and let Zn be the set of all n-tuples of nonnegative integers (i.e., the multi-indices of length n). If a = (a, ...I, an) E Zn+, we set, as usual, lal =a,+ + -+a, and a! =a1! .. !. For a,ficZ+ wewrite a + f = (a1 + f1j ... , an + fin), and a < fi whenever aci < /i (i = 1, ... , n). Let H be a Hilbert space, and let 2(H) be the algebra of all bounded linear operators on H. An element T = (T,, ... , T,) E 2(H)n such that T,, ... , T, mutually commute will be designated as a commuting multioperator (briefly, a c.m.). Let T E Y(H)n be a c.m. We define the operator MT: Y(H) -* J(H) by the formula

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Standard models for some commuting multioperators

There are a number of spectra studied in literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semi-regular or essentially semi-regular). I. Basic properties of regularities The axiomatic theory of spectrum was introduced by W. Żelazko [21], see also S lodkowski and Żelazko [17]. He gave a classification of various types of spectra defined for commuting n-tuples of elements of a Banach algebra. The most important notion is that of subspectrum. All algebras in this paper are complex and unital. Denote by Inv(A) the set of all invertible elements in a Banach algebra A and by σ(a) = {λ ∈ C, a− λ / ∈ Inv(A)} the ordinary spectrum of an element a ∈ A. The spectral radius of a ∈ A will be denoted by r(a). Definition 1.1. Let A be a Banach algebra. A subspectrum σ in A is a mapping which assigns to every n-tuple (a1, . . . , an) of mutually commuting elements of A a non-empty compact subset σ(a1, . . . , an) ⊂ C such that (1) σ(a1, . . . , an) ⊂ σ(a1)× · · · × σ(an), (2) σ(p(a1, . . . , an)) = p(σ(a1, . . . , an)) for every commuting a1, . . . , an ∈ A and every polynomial mapping p = (p1, . . . , pm) : C → C. This notion has proved to be quite useful since it includes for example the left (right) spectrum, the left (right) approximate point spectrum, the Harte (= the union of the left and right) spectrum, the Taylor spectrum and various essential spectra. However, there are also many examples of spectrum, usually defined only for single elements of A, which are not covered by the axiomatic theory of Żelazko. The aim of this paper is to give an axiomatic description of such spectra. Instead of describing a spectrum, it is possible to describe equivalently the set of regular elements. Definition 1.2. Let A be a Banach algebra. A non-empty subset R of A is called a regularity if (1) if a ∈ A and n ∈ N then a ∈ A ⇔ a ∈ A, (2) if a, b, c, d are mutually commuting elements of A and ac + bd = 1A, then ab ∈ R ⇔ a ∈ R and b ∈ R.

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On the axiomatic theory of spectrum II

Let T  ∈  B(X) be a hypercyclic operator and λ a complex number of modulus 1. Then λ T is hypercyclic and has the same set of hypercyclic vectors as T. A version of this results gives for a wide class of supercyclic operators that x ∈ X is supercyclic for T if and only if the set {tT
 n
 x  :  t > 0, n = 0, 1, ...} is dense in X. This gives answers to several questions studied in the literature.

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Rotations of Hypercyclic and Supercyclic Operators

http://www.math.cas.cz/fichier/preprints/IM_20130423133457_98.pdf

Vladimír Müller

Papers

Spectral Theory of Linear Operators: And Spectral Systems in Banach Algebras

Standard models for some commuting multioperators

On the axiomatic theory of spectrum II

Rotations of Hypercyclic and Supercyclic Operators

Distributional chaos for linear operators