1Kernels of bounded operators on the classical transfinite Banach sequence spaces

In this paper, we consider linear skew-product semiflows on bundles of Banach fibers over a locally compact metric space. Our aim is to give discrete-time theorems for the existence of global and pointwise continuous-time dichotomies with no invariant unstable manifolds. We involve here a concept of exponential dichotomy for skew-product semiflows weaker than the concept used by Sacker and Sell (J Differ Equ 15:429–458, 1974; 22:478–522, 1976) and Magalhaes (in: Chow, Hale (eds) Dynamics of infinite dimensional systems, NATO Advanced Science Institutes Series F: Computer and Systems Sciences, Springer, New York, vol 37, pp 161–168, 1987); our definition (of no past exponential dichotomy) follows roughly the definition given by Chow and Leiva (Proc Am Math Soc 124:1071–1081, 1996) in the sense that we allow the unstable subspace to have infinite dimension. The main improvement is that we go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable subspace is invariant under the cocycle). Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the variational homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the above condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p-summable spaces, sequence Orlicz spaces, etc.). Since we use a discrete-time technique we are not forced to require any continuity or measurability hypotheses on the trajectories of the exponentially bounded cocycle. Also, it is worth to mention that from discrete-time conditions we get informations about the continuous-time behavior of the solutions of differential variational equations.

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X n X_n , the completion of c 00 c_{00} relatively to the norm \[ ‖ x ‖ n = sup { ∑ i ∈ F | x i | , with , F of cardinality at most n } . \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X n X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T [ θ , A n ] T[\theta , A_n] ( 0 > θ > 1 0>\theta >1 ) associated with the regular family A n A_n , comprised of all subsets of N \mathbb {N} with at most n n elements. For this space, we characterize the extreme points and show that for θ ≤ 1 n \theta \leq \frac {1}{n} , T [ θ , A n ] T[\theta , A_n] supports standard isometries. We also derive the form for the surjective isometries of T [ θ , A 3 ] T[\theta , A_3] .

The standard delay equations are newly studied in the context of classical separable Banach Sequence Spaces. As a classical solution is shown to exist, the associated semigroup and its infinitesimal generator are found, and some important properties of those operators are proven, including some spectral properties. As an application, it is shown how can these results be used to characterize the constrained null-controllability.

In this research, we introduce the regular Copper Lucas matrix operator, which is based on the Copper Lucas sequence. We investigate the sequence spaces c 0 (Γ) and c (Γ), as well as for 1 ≤ p ≤ ∞ , all of which are linked to the newly defined regular Copper Lucas matrix Γ. Our study explores various properties and inclusion relationships among these spaces, establishes a Schauder basis, and α ‐, β ‐, and γ ‐ duals. In addition, we characterize the connections between the newly defined matrix classes and classical sequence spaces. We also examine the compactness of matrix operators within these associated sequence spaces and provide results related to specific operator ideals. Finally, we investigate the geometric properties of the associated sequence spaces.

In this work, we introduce two sequence spaces c 0 λ ( G m ) and c λ ( G m ) generated by the composition of m th order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their α-, β- and γ-duals. Lastly, we characterize some matrix classes related to those spaces. MSC:40C05, 40H05, 46B45.

Motivated by $\Phi$-summability, novel matrix operators are produced by the aid of fascinating arithmetical functions. One of them is the Riesz-Jordan totient matrix operator, introduced very recently. The primary purpose of this study is to use this matrix to obtain new spaces as the matrix domain of it in some classical sequence spaces. After presenting basic topological properties and identifying dual spaces, the characterization of certain matrix transformations is studied, defined on the resulting spaces. In the final section, the compactness of the operators on the domain of Jordan totient type matrices in the space of null sequences is researched.

We find the best constants in inequalities relating the standard norm, the dual norm, and the norm∥x∥(p,s):=inf⁡{∑k∥x(k)∥p,s}, where the infimum is taken over all finite representationsx=∑kx(k)in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces mφ,p=mφ,p(Zd). They are natural generalisations of the classical Morrey sequence spaces mu,p, 0<p≤u<∞, which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.

Nonlinear classification of Banach spaces based on geometric structure spaces

BSTRACT. In this paper, we determine the pα-duals of certain newly introduced difference se- quence spaces, namely ℓ∞(u, △m, q), c0(u, △m, q), and c(u, △m, q). Furthermore, we investigate v v v their topological properties and establish the conditions under which these spaces are perfect. 1. Introduction and preliminaries Assume that ω denotes the space of all sequences. Throughout this chapter, we adopt the notation q = (qk) to denote a sequence of positive real numbers. Some generalized sequence spaces, such as ℓ∞(q), c(q), and c0(q), have been studied by various authors [9–11]. Ahmad and Mursaleen [1] introduced the concept of some new generalized difference sequence spaces, defined as follows: △ℓ∞(q) = {x ∈ ω : △x ∈ ℓ∞(q)}. △c(q) = {x ∈ ω : △x ∈ c(q)}. △c0(q) = {x ∈ ω : △x ∈ c0(q)}. Where △xk = xk − xk+1 . After that, Et and Bas¸arer [8] defined these difference sequence spaces in a more general form by introducing the operator △mon these spaces.

In the present article, we define certain a set of new classes of sequence spaces by using Ces\`{a}ro mean and difference operator $\Delta^r\;\;r\in\mathbb{N}_0=\{0,1,2,3,\dots\}$. Also, we study the topological structures of the defined classes and determine their $\alpha-,\beta-$ and $\gamma-$ duals. Matrix transformations of given classes with their basic sequence spaces are characterized.

In this note we study the lineability and algebrability problems for certain classes of sequence spaces and series. In particular, we extend some algebrability results to the context of p-adic fields. We also prove that the subset of nonconvergent sequences of ℓ∞(ℝ) whose set of accumulation points is not a Cantor-type set is strongly 𝔠-algebrable.

This paper examines tropical recurrent sequences associated with Somos sequences. The classical Somos sequences have applications in the theory of elliptic curves. Due to the Laurent character of the classical sequences, a pattern can be inferred between the classical sequences and their tropical counterparts. The greatest interest is the increase in the dimension of the solution space of tropical sequences depending on the length of the final sequence. For a set of tropical sequences described by tropical recurrence relations, D.Yu.Grigoriev put forward a hypothesis about the stabilization of the maximum dimensions of the components of the corresponding tropical prevarieties. This hypothesis has been proven for tropical linear recurrent sequences. As part of this work, for tropical recurrent sequences associated with the sequences Somos-4 and Somos-5, the corresponding tropical prevarieties were investigated using the Gfan package in order to test Grigoriev's hypothesis.

This chapter is devoted to the classical sequence spaces and function spaces. The sequence spaces are defined explicitly but the function spaces require some measure theoretic machinery to be developed. The approach we take avoids the technicalities of measure theory and instead uses the metric formalism already at hand in order to present the function spaces both rigorously and comprehensibly. Of course, the measure theoretic techniques are of great importance. The aim is to quickly equip the reader with familiarity with these important spaces and pave the future way into measure theoretic analyses.

We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces.