On composition operators of Fibonacci matrix and applications of Hausdorff measure of noncompactness

The aim of the paper is introduced the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right).$ Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of new spaces and also construct the basis for the space $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$Finally we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators by applying the Hausdorff measure of noncompactness, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\},$ and $1\leq p<\infty.$

Capacity and the Corresponding Heat Semigroup Characterization from Dunkl-Bounded Variation

In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation.

Superdifferential Analysis of the Takagi-Van Der Waerden Functions

In this work we completely describe the superdifferential of the Takagi-Van der Waerden functions and, as a consequence, the local maxima of these functions are characterized. Regarding the set of points where the superdifferential is not empty, we calculate its Hausdorff dimension as well as its corresponding Hausdorff measure. To do so, for any even integer greater than or equal to two we determine the 1/2-dimensional Hausdorff measure of the set of points where Takagi-Van der Waerden functions attain their global maximum.

On rectifiable measures in Carnot groups: representation

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

Some Compact Operators on the Hahn Space

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space $h_{d}$, where $d$ is an unbounded monotone increasing sequence of positive real numbers, into the spaces $[c_{0}]$, $[c]$ and $[c_{\infty}]$ of sequences that are strongly convergent to zero, strongly convergent and strongly bounded. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ into $[c]$, and identities for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ to $[c_{0}]$, and use these results to characterise the classes of compact operators from $h_{d}$ to $[c]$ and $[c_{0}]$.

Banach spaces of absolutely k-summable series

In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.