Enlarge Greedy Sums in Greedy-Type Properties by Different Factors

It was previously known that the almost greedy (AG) property essentially remains the same when we enlarge greedy sums in the classical definition by a factor $\lambda \geqslant 1$. The present paper shows that if instead, we enlarge greedy sums in a reformulation of the AG property, we obtain a weaker one. However, the new property is essentially independent of the enlarging factor $\lambda$ once $\lambda > 1$. In contrast, we observe a continuum of partially greedy-like properties by varying $\lambda\in [1,\infty)$. Last but not least, under a threshold for $\lambda$, we characterize the isometric version of the weakened AG property. Specifically, the characterization holds if and only if $\lambda\in [1, 2]$.