On nonlinear translation-varying operators

In this work nonlinear translation-varying operators are analyzed and represented by means of a generalized impulse response. This is the response of the transpose operator to the family of shifted impulse functionals. Continuous operators from a topological vector space into the space of functions on Rⁿ, as well as $\text{A}$-bounded operators, are investigated.     

The Higher Spin Laplace Operator

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as well as a second-order analogue of the Rarita-Schwinger operator. To construct these operators, we will use the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the orthogonal group can be realised in terms of polynomials satisfying a system of differential equations. As a consequence, the functions on which this particular class of operators act are functions taking values in the space of harmonics homogeneous of degree k. We prove the ellipticity of these operators and use this to investigate their kernel, focusing on polynomial solutions. Finally, we will also construct the fundamental solution using the theory of Riesz potentials.     

Multiplication Operators Between the Lipschitz Space and the Space of Bounded Functions on a Tree

In this paper, we characterize the bounded and the compact multiplication operators between the space of bounded functions on the set of vertices of a rooted infinite tree T and the Banach space of complex-valued Lipschitz functions on T. We also determine the operator norm and the essential norm for the bounded multiplication operators between these spaces and show that there are no isometries among such operators.     

Subspace hypercyclicity

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity is interesting, including a nontrivial subspace-hypercyclic operator that is not hypercyclic. There is a Kitai-like criterion that implies subspace-hypercyclicity and although the spectrum of a subspace-hypercyclic operator must intersect the unit circle, not every component of the spectrum will do so. We show that, like hypercyclicity, subspace-hypercyclicity is a strictly infinite-dimensional phenomenon. Additionally, compact or hyponormal operators can never be subspace-hypercyclic.     

New Studies on the Fock Space Associated to the Generalized Airy Operator and Applications

In this work, we introduce the Fock space \(F_\nu (\mathbb {C})\) associated to the Airy operator \(L_\nu \), and we establish Heisenberg-type uncertainty principle for this space. Next, we study the Toeplitz operators, the Hankel operators and the translation operators on this space. Furthermore, we give an application of the theory of extremal function and reproducing kernel of Hilbert space, to establish the extremal function associated to a bounded linear operator \(T{:}\,F_\nu (\mathbb {C})\rightarrow H\), where H be a Hilbert space. Finally, we come up with some results regarding the extremal functions, when T is the difference operator and the Dunkl-difference operator, respectively.     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.     

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.     

Norm-attaining composition operators on the Bloch spaces

We prove that every composition operator C? on the Bloch space (modulo constant functions) attains its norm and characterize the norm-attaining composition operators on the little Bloch space (modulo constant functions). We also identify the extremal functions for ‖C?‖ in both cases.     

Operator theory in the Hardy space over the bidisk (I)

In this paper, we identify the vector valued Hardy space with the Hardy space over the bidisk and construct a universal model for thecontractive analytic functions. We will also study some elementary properties of the submodules and show, in some cases, how the operator theoretical properties are related to the module theoretical properties. The last part focus on the study of double commutativity of compression operators.     

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L:?=?P ^???T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P ^??? of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and B. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.     

Factoring multi-sublinear maps

Using Rademacher type, maximal estimates are established for k-sublinear operators with values in the space of measurable functions. Maurey–Nikishin factorization implies that such operators factor through a weak-type Lebesgue space. This extends known results for sublinear operators and improves some results for bilinear operators. For example, any continuous bilinear operator from a product of type 2 spaces into the space of measurable functions factors through a Banach space. Also included are applications for multilinear translation invariant operators.     

Linear bounds for Calderón–Zygmund operators with even kernel on UMD spaces

It is well-known that several classical results about Calderón–Zygmund singular integral operators can be extended to X-valued functions if and only if the Banach space X has the UMD property. The dependence of the norm of an X-valued Calderón–Zygmund operator on the UMD constant of the space X is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calderón–Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.     

Compact integration operators for Fréchet-space-valued measures

Many operators in Banach spaces occur as the integration operator of a suitable vector measure; their compactness is completely described in [19]. However, many important spaces X in analysis (and integration operators in such spaces) do not fall into this scheme because X is not normable. Characterizing the compactness of integration operators in this setting is the aim of this note. The methods and techniques employed are quite different to the Banach space arguments used in [19].     

Iterative Einschliessung von Lösungen Nichtlinearer Operatorgleichungen und Bestimmung von Startnäherungen

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.     

Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from operator theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but some of our results have purely algebraic formulation. We make use of Xia's theory of subnormal operators and of the previous results by the author in this direction. We also correct (in Section 5) some inaccuracies in the works of [D.V. Yakubovich, Subnormal operators of finite type I. Xia's model and real algebraic curves in C², Rev. Mat. Iberoamericana 14 (1998) 95-115; D.V. Yakubovich, Subnormal operators of finite type II. Structure theorems, Rev. Mat. Iberoamericana 14 (1998) 623-681] by the author.     

HILBERT SPACES ASSOCIATED WITH SECOND ORDER ELLIPTIC OPERATORS AND APPLICATIONS TO BOUNDARY VALUE PROBLEMS AND VARIATIONAL INEQUALITIES

Abstract

Boundary value problems and variational inequalities, associated with second order elliptic operators, will be studied in a Hilbert space framework. In this space, functions will have (at least) locally square integrable derivatives of order up to two. Also the conormal derivative, extended by continuity, will be square integrable on the boundary of the region considered. Criteria for approximating elements of the Hilbert space by smooth functions will be given and thus closed convex sets, associated with inequalities on the boundary, exist.

The idea of the present approach originated from the method suggested by Lions and Magenes, for putting some regular elliptic problems in the variational setting. The differential equation is multiplied by Qv, with Q some operator and v a function and the result is integrated as required.     

Approximation of the differentiation operator by linear bounded operators on the class of twice differentiable functions in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(0,∞)

We give an upper estimate for the value of the best approximation of the (firstorder) differentiation operator by linear bounded operators on the class of twice differentiable functions in the space L ₂(0,∞). This upper estimate is close to a known lower estimate and improves previously known upper estimates. To prove the upper estimate, we consider a specific family of operators; in this family, we choose an operator that provides the least estimate for the value of the best approximation.     

On a calculus for a generalized scalar operator

Let $\text{X}$ be a Banach space; S and T bounded scalar-type operators in $\text{X}$. Define Δ on the space of bounded operators on $\text{X}$ by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on $\text{X}$.     

Common hypercyclic vectors for the conjugate class of a hypercyclic operator

Given a separable, infinite dimensional Hilbert space, it was recently shown by the authors that there is a path of chaotic operators, which is dense in the operator algebra with the strong operator topology, and along which every operator has the exact same dense Gδ set of hypercyclic vectors. In the present work, we show that the conjugate set of any hypercyclic operator on a separable, infinite dimensional Banach space always contains a path of operators which is dense with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense Gδ set. As a corollary, the hypercyclic operators on such a Banach space form a connected subset of the operator algebra with the strong operator topology.