Abstract: We introduce several notions of linear dynamics for multivalued linearoperators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linearoperators to MLO’s is also considered.
We will not use the theory of C*-algebras in general, in fact we will use just the C*- algebras shown in example 2.4. We will define an operator f (T) for any continuous function f using an homomorphism between C*-algebras. We will construct a map from the set of all continuous functions on a compact set to the set of all bounded linearoperators on a Hilbert space. We will construct this homomorphism as an extension of the definition 2.1 using the Weierstrass Theorem.
ABSTRAC T . In [ Na-4 } we presen ted an abstract framework in which the spec trum of a linear operator can be compu ted through a characteristic equation. In tIle present note this is combined with the Perron - Frobenius spectral theory for positive operators in order to obtain simple stabili ty criteria for the solu tions of
R (T ) + N (T d ) is of ﬁnite codimension. Moreover, if T is B-Fredholm then ind(T ) = dim N (T ) ∩R (T d ) − codim R (T) + N (T d ). Based on this characterization of B-Fredholm bounded operators, we introduce the class of B-Fredholm closed linearoperators acting on a Hilbert space H and study its properties. Mainly, we prove that an operator T ∈ C (H ) densely deﬁned on H is a B-Fredholm operator if and only if T = T 0 ⊕ T 1 , where T 0 is a
Throughout this paper let B(H), F(H), K(H), denote, respectively, the algebra of bounded linearoperators, the ideal of finite rank operators and the ideal of compact operators acting on an infinite dimensional separable Hilbert space H. If T ∈ B(H) we shall write ker(T ) and R(T ) (or ran(T )) for the null space and range of T , respectively. Also, let α(T ) := dim ker(T ), β(T ) := co dim R(T ), and let σ(T ), σ a (T ), σ p (T ) denote the spectrum, approximate point spectrum and point
Once the basic properties are established, the second step was to study the boundedness of the Hardy–Littlewood maximal operator and the fractional Hardy– Littlewood maximal operator on the variable Lebesgue spaces; see Theorem 1 and Theorem 2 below and references therein. With these tools one can then study other operators such as convolution operators, singular integral operators, fractional type operators, and Riesz potentials. Several results about these operators can be found in the papers [1, 2, 7, 10, 17, 18, 20, 21].
It is well known that B´ ezier curves are the parametric curves used in computer graphics and designs. In vector graphics they are used to model smooth curves and also used in animation designs. Zeng and Piriou  pioneered the study of B´ ezier variants of Bernstein operators. The papers by other researchers (e.g., [3, 5, 8, 13, 17]) motivate us to study further in this direction.
This paper is a summary of the talk held by the author in the conference “X Encuentro de Analistas A. Calderón”, that was celebrated in La Falda, Córdoba, in Argentina, in September 2010. Our purpose is to present a survey about Hardy spaces associated with semigroups of operators. Of course it is not possible to be ex- haustive. There exist monographs about this topic (see , , and , amongst others) and almost every day a paper where Hardy spaces appear is written. This shows the great importance of the Hardy spaces.
El servicio técnico de Linear proporciona soporte telefónico temprano por la mañana y fuera del horario habitual para su línea de productos en expansión. Los ingenieros de aplicaciones están listos para ayudarlo con las preguntas técnicas respecto de todos los productos Linear y de la instalación del equipo, la programación, el diseño del sistema, la solución de problemas y la compatibilidad entre sistemas.
Note that e(F, V) is a non-linear function of F . Hence, for the problems described in the introduction, what we want is to minimize e over all possible bundles of subspaces. To show that all these problems have indeed a (constructive) solution, we need some definitions (for details we refer the reader to ).
2. In general I try to work problems in class that are different from my notes. However, with a Linear Algebra course while I can make up the problems off the top of my head there is no guarantee that they will work out nicely or the way I want them to. So, because of that my class work will tend to follow these notes fairly close as far as worked problems go. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Also, I often don’t have time in class to work all of the problems in the notes and so you will find that some sections contain problems that weren’t worked in class due to time restrictions. 3. Sometimes questions in class will lead down paths that are not covered here. I try to
Weighted ordered weighted averaging (WOWA) and semiuninorm-based ordered weighted averaging (SUOWA) operators are two families of aggre- gation functions that simultaneously generalize weighted means and OWA operators. Both families can be obtained by using the Choquet integral with respect to normalized capacities. Therefore, they are continuous, monotonic, idempotent, compensative and homogeneous of degree 1 func- tions. Although both families fulfill good properties, there are situations where their behavior is quite different. The aim of this paper is to analyze both families of functions regarding some simple cases of weighting vec- tors, the capacities from which they are building, the weights affecting the components of each vector, and the values they return.
Abstract. This paper is essentially the second author’s lecture at the CIMPA– UNESCO Argentina School 2008, Real Analysis and its Applications. It sum- marises large parts of the three authors’ paper [MSV]. Only one proof is given. In the setting of a Euclidean space, we consider operators defined and uniformly bounded on atoms of a Hardy space H p . The question discussed is
Although both families of operators allow solving a wide range of problems, both weightings are necessary in some contexts. Some examples of these situations have been given by several authors (see, for instance, Torra [2–4], Torra and Godo [5, pages 160-161], Torra and Narukawa [6, pages 150- 151], Roy , Yager and Alajlan , and Llamazares  and the references therein) in fields as diverse as robotics, vision, fuzzy logic controllers, constraint satisfaction prob- lems, scheduling, multicriteria aggregation problems, and decision-making.
Due to the fact that the calculation of deficiency spaces is difficult for dimension greater than one, another approach to find self-adjoint extensions was proposed by Friedrichs, Kato and oth- ers. This approach considers quadratic forms associated to symmetric operators. If a quadratic form is closable it is related to a unique self-adjoint operator. We arrived at Kato’s represen- tation theorem, which allows one to obtain self-adjoint extensions of symmetric, semi-bounded operators.
In this expository article we want to recount a few recent results in the study of certain multilinear operators. Multilinear harmonic analysis is an active area of research that is still developing. We will limit ourselves to results that are, in a way, natural multilinear versions of well-known and powerful theorems in the study of linear singular integrals of Calder´ on-Zygmund type. These new results only arise after many important progresses have been done in related topics and by numerous authors. This presentation is far from being exhaustive in the sense that, for reasons of space, we will not be able to describe all existing contributions in multilinear analysis but only those most closely related to operators with variable coefficients. We will concentrate on some progresses done on a series of collabora- tions and some topics presented by the author at the 2008 CIMPA-UNESCO School on Real Analysis and its Applications. One of the focus points of the conference was precisely new aspects of the Calder´ on-Zygmund theory. We will assume that the interested reader has some familiarity with basic results about linear Calder´ on- Zygmund operators and linear pseudodifferntial operators, but refer to the book by Stein  for a comprehensive introduction. Also for brevity, we will not present the theorems in their greatest generality, but with hypotheses that simplify the narrative and still encapsulate the main mathematical aspects involved.