Mathematics

Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert C ∗ -modules. In this paper, we introduce the concept of Controlled K-operator frame for the space En d ∗ A (H) of all adjointable operators on a Hilbert A -module H and we establish some results.

It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper.

Given a frequency λ , we study general Dirichlet series ??a n e ??λ n s . First, we give a new condition on λ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in the right half-plane converges uniformly in this half-plane, improving classical results of Bohr and Landau. Then, following recent works of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series. We get general results on almost sure convergence which have an harmonic analysis flavour. Nevertheless, we also exhibit examples showing that it seems hard to get general results on these spaces as spaces of holomorphic functions.

In this paper, we prove the strong convergence theorems for nearly nonexpansive mappings, using the modified Picard-Mann hybrid iteration process in the context of uniformly convex Banach space.

For vector lattices E and F , where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators L ob (E,F) from E into F . Using this, it follows that L ob (E,F) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and-when applicable-convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on L ob (E,F) . Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms on a Banach lattice, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We also show that, in contrast to general order bounded operators, the orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and -- when applicable -- convergence in the Hausdorff uo-Lebesgue topology as well.

We introduce and investigate the orbit-closed C -numerical range, a natural modification of the C -numerical range of an operator introduced for C trace-class by Dirr and vom Ende. Our orbit-closed C -numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when C is finite rank. Since Dirr and vom Ende's results concerning the C -numerical range depend only on its closure, our orbit-closed C -numerical range inherits these properties, but we also establish more. For C selfadjoint, Dirr and vom Ende were only able to prove that the closure of their C -numerical range is convex, and asked whether it is convex without taking the closure. We establish the convexity of the orbit-closed C -numerical range for selfadjoint C without taking the closure by providing a characterization in terms of majorization, unlocking the door to a plethora of results which generalize properties of the C -numerical range known in finite dimensions or when C has finite rank. Under rather special hypotheses on the operators, we also show the C -numerical range is convex, thereby providing a partial answer to the question posed by Dirr and vom Ende.

In this paper, for a locally compact commutative hypergroup K and for a pair ( Φ 1 , Φ 2 ) of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of K, for the convolution f?�g to exist a.e., where f and g are arbitrary elements of Orlicz spaces L Φ 1 (K) and L Φ 2 (K) , respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from L 1 w (K) into L Φ w (K) .

We establish convolution inequalities for Besov spaces B s p,q and Triebel--Lizorkin spaces F s p,q . As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces A s p,q , A?�{B,F} . Our results apply to a wide class of convolution semigroups including the Gau?--Weierstra? semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian (?��?) m , and we can derive various caloric smoothing estimates.

In this paper we aim to generalize results obtained in the framework of fractional calculus by the way of reformulating them in terms of operator theory. In its own turn, the achieved generalization allows us to spread the obtained technique on practical problems that connected with various physical - chemical processes.

In this note we correct a paper by D. Kang ("On the Mazur-Ulam theorem in non-Archimedean fuzzy anti-2-normed spaces", Filomat, 2017). The research in that paper applies to what the author calls strictly convex spaces. Nevertheless, we prove that this notion is void: there is no single space that satisfies the definition.