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.

Cyclicity of Special Operators on a BK with AK Space

Let Ω be a complex domain and let F be a reflexive BK space with AK such that F^⊂H(Ω) and the functional of evaluation at λ is bounded for all λ∈Ω. We will investigate the cyclicity for the adjoint of a weighted composition operator acting on F^.

SIP Xd-frames and their perturbations in uniformly convex Banach spaces

In this paper, we introduce the definitions of SIP-I and SIP-II Xd-frames in a uniformly convex, separable Banach space X with respect to a BK-space Xd (here SIP represents semi-inner product), both of them are defined as sequence of elements in X. We characterize SIP-I and SIP-II Xd-frames in terms of the corresponding synthesis and analysis operators, respectively, then we consider perturbations for both of them. Furthermore, we also introduce the definitions of SIP Banach frames and SIP atomic decompositions. Under certain assumptions, we establish the relationship between SIP Banach frames and SIP atomic decompositions, and therefore obtain reconstruction formulas for every element in X and X* by using a pair of SIP-I and SIP-II Xd-frames for X. Finally, we discuss perturbations of SIP Banach frames and SIP atomic decompositions.

FRÉCHET FRAMES, GENERAL DEFINITION AND EXPANSIONS

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, ‖ ⋅ ‖1 ≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XF and its dual [Formula: see text], as well of some of the generating spaces of XF with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations.

Some Topological and Geometrical Properties of a New Difference Sequence Space

We introduce the new difference sequence space . Further, it is proved that the space is the BK-space including the space , which is the space of sequences of pbounded variation. We also show that the spaces , and are linearly isomorphic for . Furthermore, the basis and the , and duals of the space are determined. We devote the final section of the paper to examine some geometric properties of the space .

On some new sequence spaces of non-absolute type related to the spaces ℓp and ℓ∞ I

In the present paper, we introduce the sequence space l?p of non-absolute type and prove that the spaces ??p and lp are linearly isomorphic for 0 < p ? ?. Further, we show that ??p is a p-normed space and a BK-space in the cases of 0 < p < 1 and 1 ? p ? ?, respectively. Furthermore, we derive some inclusion relations concerning the space ??p. Finally, we construct the basis for the space ??p, where 1 ? p < ?.

Onβ-dual of vector-valued sequence spaces of Maddox

Theβ-dual of a vector-valued sequence space is defined and studied. We show that if anX-valued sequence spaceEis a BK-space having AK property, then the dual space ofEand itsβ-dual are isometrically isomorphic. We also give characterizations ofβ-dual of vector-valued sequence spaces of Maddoxℓ(X,p),ℓ∞(X,p),c0(X,p), andc(X,p).