The superposition operator in ideal spaces

Problems of solving different linear difference equation is given to study the properties of the matrix operators in various functional spaces. One of the important problems of functional analysis is to establish criteria of boundedness of the linear operators in functional spaces. Question of the boundedness of matrix operators in sequence spaces is a classic problem of functional analysis and there are many unsolved problems in it. For example, in the general case it is impossible to establish the boundedness of the matrix operator in the spaces of sequences by the given matrix. Therefore, various classes of matrix operators are considered for which the criteria of their boundedness are known. Due to the variety of encountered problems in practice, it is necessary to have various alternative criteria for the boundedness of matrix operators. In this paper, we establish a new alternative criterion for the boundedness of one class of matrix operators.

In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014).

The tensor and exterior squares of a completely continuous nonnegative linear operator A acting in the ideal space X(Ω) are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square (A ⊗ A) M in the terms of the spectrum of the initial operator A is proved. The existence of the second (according to the module) positive eigenvalue λ2, or a pair of complex conjugate eigenvalues of a completely continuous non-negative operator A is proved under the additional condition, that its exterior square (A ∧ A) M is also nonnegative.

Geophysical Inverse Theory and Regularization Problems

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections onK(X, Y) for Banach spacesX, Y. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result forL(X, Y) thereby proving an analogue of the result from [9] forL p (1 <p ≠ 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

Motivated by the importance of conjugate spaces indicated in earlier sections, we devote the bulk of this final chapter to some further considerations regarding such spaces. We begin with the famous Riesz-Kakutani characterization of C(Ω, ℝ)* as the space of regular signed Borel measures on Ω. After giving some applications of this theorem we proceed to some characterizations of general conjugate spaces, and use these to exhibit some new conjugate spaces (spaces of operators and Lipschitz functions). The fact that certain spaces of operators are conjugate spaces has some interesting implications for optimization theory as we shall see. We shall also establish an isomorphism between certain spaces of Lipschitz functions and certain spaces of L00 type. A particular consequence of this is an example of a pair of Banach spaces (namely, ℓ1(ϰ0) and L1([0, 1])) which fail to be isomorphic, yet whose conjugate spaces are isomorphic.

In this paper we present a necessary condition for an autonomous superposition operator to act in the space of functions of Waterman-Shiba bounded variation. We also show that if a (general) superposition operator applies such space into itself and it is uniformly bounded, then its generating function satisfies a weak Matkowski condition.

This book is a self-contained account of knowledge of the theory of nonlinear superposition operators: a generalization of the notion of functions. The theory developed here is applicable to operators in a wide variety of function spaces, and it is here that the modern theory diverges from classical nonlinear analysis. The purpose of this book is to collect the basic facts about the superposition operator, to present the main ideas which are useful in studying its properties and to provide a comparison of its behaviour in different function spaces. Some applications are also considered, for example to control theory and optimization. Much of the work here has only appeared before in research literature, which itself is catalogued in detail here.

Given a Banach lattice L, the space of lattice Lipschitz operators on L has been introduced as a natural Lipschitz generalization of the linear notions of diagonal operator and multiplication operator on Banach function lattices. It is a particular space of superposition operators on Banach lattices. Motivated by certain procedures in Reinforcement Learning based on McShane–Whitney extensions of Lipschitz maps, this class has proven to be useful also in the classical context of Mathematical Analysis. In this paper, we discuss the properties of such operators when defined on spaces of continuous functions, focusing attention on the functional bounds for the pointwise Lipschitz inequalities defining the lattice Lipschitz operators, the representation theorems for these operators as vector-valued functions, and the corresponding dual spaces. Finally, and with possible applications in Artificial Intelligence in mind, we provide a McShane–Whitney extension theorem for these operators.

The starting point for this investigation was an attempt to generalize the well known theorem of Silverman and Toeplitz [25](1) on regularity of a sequence-to-sequence transformation. This theorem may be stated as follows: If a transformation of sequences { t, } of real numbers to sequences { Sm } is defined from a matrix { amn }, m, n = 1, 2, * , by the equations Sm =ZEn amntn, the transformation is regular-that is, is defined everywhere and takes every convergent sequence { t, } into another convergent sequence with the same limit-if and only if the matrix { amn } satisfies the conditions (a) limm dn 'tamn = 1, (b) limm amn = 0 for each n, and (c) there is a K such that En I amn, < K for every m. In the special case under consideration, the fact that regularity implies condition (c) (the non-trivial part of the proof) can be derived from a theorem of Banach [3, p. 80, Theorem 5 ]: If A and B are Banach spaces, and if U, n = 1, 2, * * , are. linear operators on A to B, such that lim supn || Un(a)ll < o? for each a in A, then lim supn 11 U711 < oo If the sequence of integers is replaced by a directed set X, it is known that A, B, X, and U. can be chosen for which the similar statement relating lim supx || Ux(a)ll and lim supx || Ux,l is false; sections 1-3 of this paper consider these cases in an attempt to solve the problem of boundedness: Characterize those Banach spaces A and B, and directed sets X such that choosing the linear operators U, on A to B so that lim supx || U,(a)ll < oo for each a in A implies that lim supx || Ux|| < oo. Section 1 is a review of pertinent facts about directed sets and convergence (mostly due to Moore and Smith [19], G. Birkhoff [5], and Tukey [27]). Section 2 studies the relations among three topologies in the space of operators on A to B. In ?3 the problem of boundedness is studied but not completely solved. The second part of the paper is concerned with certain special operators on some function spaces. In ?4 the space is that of the totally measurable functions on a set Y to a Banach space B; a class of operators on this space is defined in terms of additive, real-valued set-functions and the relations among various topologies in this set of operators is given; this is used in ?6 to give a general form to a theorem of Vulich [28]. In ?5 the functions studied are the measurable functions on Y to B; the operations on this space are defined in terms of completely additive, limited, set-functions whose values are transfor-

Superposition operators on spaces of analytic functions

In this paper one considers some general theorems of the theory of Hankel and Toeplitz operators in spaces of analytic functions. Under natural restrictions on the spaces X, it is shown that the symbols of the Toeplitz operators, acting in X, are bounded. One describes completely the symbols of the Hankel and Toeplitz operators, acting from Hp into ¯Hq (into Hq) for 0 < p, q < ∞.

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

We consider the topological space of all composition operators, acting on certain Hilbert spaces of holomorphic functions on the unit disc, in the uniform operator topology. A sufficient condition is given for the component of a composition operator to be a singleton. A necessary condition is given for one composition operator to lie in the component of another. In addition, we prove analogous results for the component of the image of a composition operator in the Calkin algebra. Finally, we obtain some related results on the essential norm of a linear combination of composition operators.

Learning reliable embeddings requires large data and is not trivial to be adapted to specific tasks due to certain constraints. For queries, the lack of sufficient context can prohibit learning quality representations and thus effective query understanding. We propose to project embeddings from a source space learned with natural language into a target space on queries. The projection function is learned via an overlap vocabulary set shared by both source and target spaces. Experimental results show that both linear and nonlinear embedding projections can help query intent classification and query slot tagging, even when the amount of data used for learning the projection is limited.