Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

An Examination of Per Capita Income Convergence in Emerging Market Economies

The study intends to examine the convergence of per capita income in emerging market economies (EMEs) toward a steady state for the post reform period (1999–2019). Cross-sectional regression analysis has been performed for unconditional convergence and a panel data regression to find the conditional convergence in EMEs. Sigma convergence has been applied to find the dispersion of income level in EMEs. In addition, to find the impact of global financial crisis on the convergence process of EMEs, unit root test with one structural break has been applied. The findings indicate that there exists unconditional convergence among EMEs toward a common steady state. Further, the results show a significant role of all control variables except education in the growth process but prove the absence of conditional convergence in selected EMEs. The results of sigma convergence find that the dispersion of per capita income is declining in EMEs, showing the sign of sigma convergence in EMEs. However, this study provides further scope to examine per capita income convergence among EMEs by including other variables and their effect on the convergence process of EMEs.

ON THE UNCONDITIONAL CONVERGENCE OF FABER--SCHAUDER SERIES IN $L^{1}$

In this paper we proved that the Faber--Schauder functions form an unconditional representation system for $L^1$.

Rethinking Unconditional Convergence in Manufacturing in the Age of New Technologies

We revisit unconditional convergence within manufacturing with a focus on differences in technology intensity across industries. For Latin American and Sub-Saharan African economies, we observe that low-technology and medium-technology intensive industries experience a significantly slower convergence in comparison to high-technology intensive ones. In contrast, we find no evidence of a significant differential for low-technology industries’ convergence in Asian economies, and if anything, we see that medium-technology intensive sectors experience a faster convergence than high-technology industries. In developed economies, we observe that while low-technology industries experience a slightly slower convergence, medium-technology industries converge at similar rates to high-technology industries. We also find that these differences emerge during the period of increased global integration, which exposed developing economies to increased competition both from advanced markets and fast industrializers within the developing world. Finally, we show that differential convergence patterns are stronger after the peak of manufacturing employment share has been reached. We discuss the implications of these trends for the future of development policy making.