Lipschitz Operators Research Articles

Eccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells space, along with the duality between M and the metric dual space M# defined by the real-valued Lipschitz functions on M. However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of M# for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.

Lipschitz Operators Associated with Weakly $p$-Compact and Unconditionally $p$-Compact Sets

The study introduces different categories of Lipschitz operators linked with weakly $p$-compact and unconditionally $p$-compact sets. It explores some properties of these operator classes derived from linear operators associated with these sets and examines their interconnections. Additionally, it denotes that these classes are extensions of the related linear operators. Moreover, the study evaluates the concept of majorization by scrutinizing both newly obtained and pre-existing results and draws some conclusions based on these findings. The primary method used to obtain the results in the study is the linearization of Lipschitz operators through the Lipschitz-free space constructed over a pointed metric space.

A Theory for Interpolation of Metric Spaces

In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without any algebraic structure required. Furthermore, this interpolation method for metric spaces when applied to normed spaces is equivalent to the K-method, which has been widely studied in the literature. As an application, we interpolate Fréchet sequence spaces.

Absolutely Fuzzy Lipschitz p-summing Maps between Fuzzy Pointed Metric Spaces

Motivated by Lipschitz ideals in the conventional (crisp) theory, we are constructing a new Lipschitz ideal in fuzzy theory. We introduce the notion of fuzzy Lipschitz ideals and give some elementary illustrations. The class of absolutely fuzzy Lipschitz -summing maps between arbitrary fuzzy pointed metric spaces is a significant category of fuzzy Lipschitz ideals. It is a logical extension of the concept of absolutely (crisp) Lipschitz p-summing maps between arbitrary pointed metric spaces, as established by Farmer Jeffrey and William Johnson. We establish that the fuzzy Lipschitz norm of the previously specified concept is a fuzzy real number. We demonstrate that a complete fuzzy normed fuzzy operator ideal is the resulting class of fuzzy Lipschitz operators between arbitrary fuzzy pointed metric spaces and complete fuzzy normed spaces. Next, we define a basic characterisation of a Lipschitz p-summing map that is completely fuzzy. By demonstrating a fuzzy variant of the nonlinear Pietsch Domination Theorem, this is accomplished. Lastly, we bring forth a few unsolved problems that we find intriguing.

On Substitutions with a Weight in the Space of Operator Lipschitz Functions

On Substitutions with a Weight in the Space of Operator Lipschitz Functions

The extension of two-Lipschitz operators

The paper deals with some further results concerning the class of two-Lipschitz operators. We prove first an isometric isomorphism identification of two-Lipschitz operators and Lipschitz operators. After defining and characterize the adjoint of two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from cartesian product of two complemented subspaces of a Banach space to the cartesian product of whole spaces. Also, we show that every two-Lipschitz functional defined on cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm, under some requirements on domaine spaces.

Approximation of Almost Diagonal Non-linear Maps by Lattice Lipschitz Operators

Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction.

Recurrence and vectors escaping to infinity for Lipschitz operators

We investigate dynamical properties of linear operators that are obtained as the linearization of Lipschitz self-maps defined on a pointed metric space. These operators are known as Lipschitz operators. More precisely, for a Lipschitz operator fˆ, we study the set of recurrent vectors and the set of vectors μ such that the sequence (‖fˆn(μ)‖)n goes to infinity. As a consequence of our results we get that there is no wild Lipschitz operator. Furthermore, several examples are presented illustrating our ideas. We highlight the cases when the underlying metric space is a connected subset of R or a subset of Zd. We end this paper studying some topological properties of the set of Lipschitz operators.

Fixed point theorems for nonexpansive and 𝒫-Lipschitzian mappings

The purpose of this paper is to present some new fixed point results for nonexpansive and [Formula: see text]-Lipschitzian operators as defined in a Banach algebra. Also we try to show that these results can be proved under some weaker conditions. An example showing the importance of our result is also included.

Optimal extensions of Lipschitz maps on metric spaces of measurable functions

We prove a factorization theorem for Lipschitz operators acting on certain subsets of metric spaces of measurable functions and with values on general metric spaces. Our results show how a Lipschitz operator can be extended to a subset of other metric space of measurable functions that satisfies the following optimality condition: it is the biggest metric space, formed by measurable functions, to which the operator can be extended preserving the Lipschitz constant. As an application, we show the coarsest metric that can be given for a metric space in which an order bounded lattice-valued-Lipschitz map is defined. Concrete examples involving the relevant space L0(μ) are given.

Extension procedures for lattice Lipschitz operators on Euclidean spaces

We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions.

Tseng’s Algorithm with Extrapolation from the past Endowed with Variable Metrics and Error Terms

In this paper, we propose a variable metric version of Tseng’s algorithm (the forward-backward-forward algorithm: FBF) combined with extrapolation from the past that includes error terms for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. The algorithm, which is a modified forward-reflected-backword method, is endowed with variable metrics and error terms. Primal-dual algorithms are also proposed for monotone inclusion problems involving compositions with linear operators. The primal-dual problem occurring in image deblurring demonstrates an application of our theoretical results.

On Operator Lipschitz Norm of the Function zn on Finite Subsets of the Unit Circle

On Operator Lipschitz Norm of the Function zn on Finite Subsets of the Unit Circle

Representation of Lipschitz Maps and Metric Coordinate Systems

Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze “Pietsch Theorem inspired factorizations" through subspaces of ℓ∞ and L1, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given.

Iterative algorithms for general variational inequalities involving relaxed Lipschitz operators

In this paper, we establish and discuss the existence of a solution of the general variational inequality : Find u ∈ H, and ω ∈ T(u) such that h(u) ∈ K and This variational inequality was studied by Verma [3] in case s ≡ 0, and g ≡ I.

Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions

In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward splitting, among other methods. At each iteration, our algorithm needs two computations of the Lipschitzian operator while the cocoercive operator is activated only once. By using product space techniques, we derive a method for solving a composite monotone primal-dual inclusions including linear operators and we apply it to solve constrained composite convex optimization problems. Finally, we apply our algorithm to a constrained total variation least-squares problem and we compare its performance with efficient methods in the literature.

Compact and weakly compact Lipschitz operators

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$ . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

Boundedness of the Hilbert transform in Lorentz spaces and applications to operator ideals

In this paper, it is investigated the optimal range of the classical Hilbert transform in Lorentz spaces of functions and its non-commutative counterparts including a triangular truncation operator in Schatten-Lorentz ideals. Some applications of obtained results to operator Lipschitz functions and commutator estimates in Schatten-Lorentz ideals of compact operators are presented.

Error estimates for DeepONets: a deep learning framework in infinite dimensions

Abstract DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ordinary differential equation, an elliptic partial differential equation (PDE) with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $\epsilon $ is argued to suffer from a ‘curse of dimensionality’ (requiring a neural networks of exponential size in $1/\epsilon $), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $\epsilon $ with neural networks of size that can grow algebraically in $1/\epsilon $).Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.

Convergence analysis of the stochastic reflected forward–backward splitting algorithm

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate $${\mathcal {O}}(log(n)/n)$$ in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex–concave saddle point problems, we derive the rate of the primal–dual gap. In particular, we also obtain $${\mathcal {O}}(1/n)$$ rate convergence of the primal–dual gap in the deterministic setting.