Abstract: This paper addresses the data-driven identification of latent representations of partially observed dynamical systems, i.e., dynamical systems for which some components are never observed, with an emphasis on forecasting applications and long-term asymptotic patterns. Whereas state-of-the-art data-driven approaches rely in general on delay embeddings and linear decompositions of the underlying operators, we introduce a framework based on the data-driven identification of an augmented state-space model using a neural-network-based representation. For a given training dataset, it amounts to jointly reconstructing the latent states and learning an ordinary differential equation representation in this space. Through numerical experiments, we demonstrate the relevance of the proposed framework with respect to state-of-the-art approaches in terms of short-term forecasting errors and long-term behavior. We further discuss how the proposed framework relates to the Koopman operator theory and Takens’ embedding theorem.

Abstract: We consider bounded operators A acting iteratively on a finite set of vectors {fi: i ∈ I} in a Hilbert space ℌ and address the problem of providing necessary and sufficient conditions for the collection of iterates {Anfi: i ∈ I, n = 0, 1, 2,...} to form a frame for the space ℌ. For normal operators A we completely solve the problem by proving a characterization theorem. Our proof incorporates techniques from different areas of mathematics, such as operator theory, spectral theory, harmonic analysis, and complex analysis in the unit disk. In the second part of the paper we drop the strong condition on A to be normal. Despite this quite general setting, we are able to prove a characterization which allows to infer many strong necessary conditions on the operator A. For example, A needs to be similar to a contraction of a very special kind. We also prove a characterization theorem for the finite-dimensional case. These results provide a theoretical solution to the so-called dynamical sampling problem where a signal f that is evolving in time through iterates of an operator A is spatially sub-sampled at various times and one seeks to reconstruct the signal f from these spatial-temporal samples.

Abstract: In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz–Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces $$W^{s,p}$$ to include the general case $$W^sL_A$$ , where A is an N-function and $$s\in (0,1)$$ . We are concerned with some qualitative properties of the space $$W^sL_A$$ (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.

Abstract: We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the B-numerical radius of $$2 \times 2$$ operator matrices where $$B = \textit{diag}(A,A)$$ , A being a positive operator. As an application of the A-numerical radius inequalities, we obtain a bound for the zeros of a polynomial which is quite a bit improvement of some famous existing bounds for the zeros of polynomials.

Abstract: We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Kothe function spaces and Riesz spaces.

Abstract: In this pages, we discuss the problem of equivalence between fractional differential and integral problems. Although the said problem was studied for ordinary derivatives, it makes some troubles in the case of fractional derivatives. This paper dealing with the question revealing the equivalence between the boundary value problems for Caputo-type fractional differential equations and the corresponding integral form. One of our main part is to determine the scope of equivalence of such problems. However, with the help of appropriate examples, we will show that even for the Holderian functions, but being off the space of absolutely continuous functions, the equivalence between the Caputo-type fractional differential problems and the corresponding integral forms can be lost. As a pursuit of this, we will show here that, the main results obtained by many researchers contained a common mathematical error in the proof of the equivalence of the boundary value problems for Caputo-type fractional differential equations and the corresponding integral forms. In this connection, we are going to slightly modify the definition of the Caputo-type fractional differential operator into a more suitable one. We will show that the modified definition is more convenient in studying the boundary value problems for Caputo-type fractional differential equations. As an application, after recalling some properties of the said (new) operator, we summarize our discussion by presenting an equivalence result for differential and integral problems for Caputo-type fractional derivatives and for the weak topology. In order to cover the full scope of this paper, we investigate the equivalence problem in case of multivalued fractional problems.

Abstract: In this paper, we define and study the notion of localization operators associated with the Weinstein–Wigner transform. We prove that they are in the trace class $$S_1$$ and give a trace formula for them. At last, we study their boundedness and compactness on $$L^p_\alpha ({\mathbb {R}}^{d+1}_+),\;1\le p\le \infty .$$

Abstract: Finding an embedding space for a linear approximation of a nonlinear dynamical system enables efficient system identification and control synthesis. The Koopman operator theory lays the foundation for identifying the nonlinear-to-linear coordinate transformations with data-driven methods. Recently, researchers have proposed to use deep neural networks as a more expressive class of basis functions for calculating the Koopman operators. These approaches, however, assume a fixed dimensional state space; they are therefore not applicable to scenarios with a variable number of objects. In this paper, we propose to learn compositional Koopman operators, using graph neural networks to encode the state into object-centric embeddings and using a block-wise linear transition matrix to regularize the shared structure across objects. The learned dynamics can quickly adapt to new environments of unknown physical parameters and produce control signals to achieve a specified goal. Our experiments on manipulating ropes and controlling soft robots show that the proposed method has better efficiency and generalization ability than existing baselines.

Abstract: In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on $$\ell ^2({\mathbb {Z}}^n)$$. In this setting, we introduce a suitable notion of a class of classical symbols which provide a class of Dixmier traceable discrete pseudo-differential operators. We also present a formula for the Dixmier trace of a Dixmier traceable discrete pseudo-differential operator by using the Connes equivalence between the Wodzicki residue and the Dixmier trace.

Abstract: In this article, we introduce an explicit parallel algorithm for finding a common element of zeros of the sum of two accretive operators and the set of fixed point of a nonexpansive mapping in the framework of Banach spaces. We prove its strong convergence under some mild conditions. Finally, we provide some applications to the main result. The results presented in this paper extend and improve the corresponding results in the literature.

Abstract: Let X be a Banach space and A, B, C, D be bounded linear operators on X. In this paper we show that the Drazin (resp. generalized Drazin) spectrum of AC coincides with the Drazin (resp. generalized Drazin) spectrum of BD under certain assumptions. Moreover, several interesting Banach space properties, such as SVEP and polaroidness, are considered.

Abstract: We introduce a convolution form, in terms of integration over the unit disc $$\mathbb {D},$$ for operators on functions f in $$H(\mathbb {D})$$ , which correspond to Taylor expansion multipliers. We demonstrate advantages of the introduced integral representation in the study of mapping properties of such operators. In particular, we prove the Young theorem for Bergman spaces in terms of integrability of the kernel of the convolution. This enables us to refer to the introduced convolutions as Hadamard–Bergman convolution. Another, more important, advantage is the study of mapping properties of a class of such operators in Holder type spaces of holomorphic functions, which in fact is hardly possible when the operator is defined just in terms of multipliers. Moreover, we show that for a class of fractional integral operators such a mapping between Holder spaces is onto. We pay a special attention to explicit integral representation of fractional integration and differentiation.

Abstract: A generalization of the products of composition, multiplication and differentiation operators is the Stevic–Sharma operator $$T_{u_1,u_2,\varphi }$$ , defined by $$T_{u_1,u_2,\varphi }f=u_1\cdot f\circ \varphi +u_2\cdot f'\circ \varphi $$ , where $$u_1,u_2,\varphi $$ are holomorphic functions on the unit disk $${\mathbb {D}}$$ in the complex plane $${\mathbb {C}}$$ and $$\varphi ({\mathbb {D}})\subset {\mathbb {D}}$$ . We are interested in the difference of Stevic–Sharma operators which has never been considered so far. In this paper, we characterize its boundedness, compactness and order boundedness between Banach spaces of holomorphic functions. As an important special case, we obtain the above characterizations of the difference of weighted composition operators. Furthermore, we show the equivalence of order boundedness and Hilbert-Schmidtness for the difference of composition operators between Hardy or weighted Bergman spaces.

Abstract: In this paper, we investigate the lower and upper bound of the relative operator $$(\alpha ,\beta )$$-entropy. We refine and improve the lower and upper bound of the relative operator entropy and generalized relative operator entropy. As a consequence of our result, the bounds of the relative operator entropy announced by Fujii and Kamei will improve.

Abstract: We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.

Abstract: Weaving frames are powerful tools in wireless sensor networks and pre-processing signals. In this paper, we introduce the concept of weaving for g-frames in Hilbert spaces. We first give some properties of weaving g-frames and present two necessary conditions in terms of frame bounds for weaving g-frames. Then we study the properties of weakly woven g-frames and give a sufficient condition for weaving g-frames. It is shown that weakly woven is equivalent to woven. Two sufficient conditions for weaving g-Riesz bases are given. And a weaving equivalent of an unconditional g-basis for weaving g-Riesz bases is considered. Finally, we present Paley–Wiener-type perturbation results for weaving g-frames.

Abstract: We give some sufficient and necessary conditions for translation operators on the weighted Orlicz spaces to be disjoint topologically transitive and disjoint topologically mixing. In particular, we show that in certain cases, operators are disjoint topologically transitive if, and only if, their direct sum is topologically transitive.

Abstract: In this paper we solve a problem posed by Bommier-Hato et al. (J Math Anal Appl 389:1086–1104, 2012) on the boundedness of the Bergman-type projections in generalized Fock spaces. It will be a consequence of two facts: a full description of the embeddings between generalized Fock–Sobolev spaces and a complete characterization of the boundedness of the above Bergman type projections between weighted $$L^p$$-spaces related to generalized Fock–Sobolev spaces.

Abstract: In this article, we study Schrodinger operators on the real line, when the external potential represents a dislocation in a periodic medium. We study how the spectrum varies with the dislocation parameter. We introduce several integer-valued indices, including the Chern number for bulk indices, and various spectral flows for edge indices. We prove that all these indices coincide, providing a proof of a bulk-edge correspondence in this case. The study is also made for dislocations in Dirac models on the real line. We prove that 0 is always an eigenvalue of such operators.

Abstract: In this paper, we investigate necessary and sufficient conditions under which compact operators between Banach lattices must be almost L-weakly compact (resp. almost M-weakly compact). Mainly, it is proved that if X is a non zero Banach space then every compact operator $$T{:}X\rightarrow E$$ (resp. $$T{:}E\rightarrow X$$) is almost L-weakly compact (resp. almost M-weakly compact) if and only if the norm on E (resp. $$E^{\prime }$$) is order continuous. Moreover, we present some interesting connections between almost L-weakly compact and L-weakly compact operators (resp. almost M-weakly compact and M-weakly compact operators).

Abstract: We provide a framework for learning of dynamical systems rooted in the concept of representations and Koopman operators. The interplay between the two leads to the full description of systems that can be represented linearly in a finite dimension, based on the properties of the Koopman operator spectrum. The geometry of state space is connected to the notion of representation, both in the linear case - where it is related to joint level sets of eigenfunctions - and in the nonlinear representation case. As shown here, even nonlinear finite-dimensional representations can be learned using the Koopman operator framework, leading to a new class of representation eigenproblems. The connection to learning using neural networks is given. An extension of the Koopman operator theory to "static" maps between different spaces is provided. The effect of the Koopman operator spectrum on Mori-Zwanzig type representations is discussed.

Abstract: We obtain a characterization for the commutant of certain weighted shift operators of higher multiplicity. The characterization is based on the realization of the operator as multiplication by monomials on weighted Bergman spaces of the unit disk.

Abstract: Columns and rows are operations for pairs of linear relations in Hilbert spaces, modelled on the corresponding notions of the componentwise sum and the usual sum of such pairs. The introduction of matrices whose entries are linear relations between underlying component spaces takes place via the row and column operations. The main purpose here is to offer an attempt to formalize the operational calculus for block matrices, whose entries are all linear relations. Each block relation generates a unique linear relation between the Cartesian products of initial and final Hilbert spaces that admits particular properties which will be characterized. Special attention is paid to the formal matrix multiplication of two blocks of linear relations and the connection to the usual product of the unique linear relations generated by them. In the present general setting these two products need not be connected to each other without some additional conditions.

Abstract: In this paper, we give necessary and sufficient conditions for the boundedness of rough Hausdorff operators on Herz, Morrey and Morrey–Herz spaces with absolutely homogeneous weights. Especially, the estimates for operator norms in each case are worked out. Moreover, we also establish the boundedness of the commutators of rough Hausdorff operators on the two weighted Morrey–Herz type spaces with their symbols belonging to Lipschitz space. By these, we obtain some new estimates for the high dimensional Hardy operator and adjoint Hardy operator.

Abstract: We prove comparison principles for nonlinear potential theories in euclidian spaces in a very straightforward manner from duality and monotonicity. We shall also show how to deduce comparison principles for nonlinear differential operators, a program seemingly different from the first. However, we shall marry these two points of view, for a wide variety of equations, under something called the correspondence principle. In potential theory one is given a constraint set F on the 2-jets of a function, and the boundary of F gives a differential equation. There are many differential operators, suitably organized around F, which give the same equation. So potential theory gives a great strengthening and simplification to the operator theory. Conversely, the set of operators associated to F can have much to say about the potential theory. An object of central interest here is that of monotonicity, which explains and unifies much of the theory. We shall always assume that the maximal monotonicity cone for a potential theory has interior. This is automatic for gradient-free equations where monotonicity is simply the standard degenerate ellipticity and properness assumptions. We show that for each such potential theory F there is an associated canonical operator, defined on the entire 2-jet space and having all the desired properties. Furthermore, comparison holds for this operator on any domain which admits a regular strictly M-subharmonic function, where M is a monotonicity subequation for F. On the operator side there is an important dichotomy into the unconstrained cases and constrained cases, where the operator must be restricted to a proper subset of 2-jet space. These two cases are best illustrated by the canonical operators and Dirichlet-Garding operators, respectively. The article gives many, many examples from pure and applied mathematics, and also from theoretical physics.

Abstract: The purpose of this paper is to present in Hilbert spaces some results for a new class of operators called semi-generalized partial isometries which include two important classes in operator theory: partial isometries and nilpotent operators. We prove some basic properties and decomposition theorems. Some spectral theorems for this class of operators are also given. Part of the results proved in this paper improve and generalize some results known for partial isometries.

Abstract: We prove that a unital, bijective linear map between absolute order unit spaces is an isometry if and only if it is absolute value preserving. We deduce that, on (unital) JB-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that a unital, bijective $$*$$ -linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) $$\hbox {C}^*$$ -algebras such maps are precisely $$\hbox {C}^*$$ -algebra isomorphisms.

Abstract: Isometries played a pivotal role in the development of operator theory, in particular with the theory of contractions and polar decompositions and has been widely studied due to its fundamental importance in the theory of stochastic processes, the intrinsic problem of modeling the general contractive operator via its isometric dilation and many other areas in applied mathematics. In this paper we present some properties of n-quasi-(m;C)-isometric operators. We show that a power of a n-quasi-(m;C)-isometric operator is again a n-quasi-(m;C)-isometric operator and some products and tens