Equicompact sets of operators defined on Banach spaces

Let $X$ and $Y$ be Banach spaces. We say that a set $\mathcal{M}\subset\mathcal{K}(X,Y)$ $(\mathcal{K}(X,Y)$ denotes the space of all compact operators from $X$ into $Y$) is equicompact if there exists a null sequence $(x_n^*)_n$ in $X^*$ such that $\Vert Tx\Vert\leq\sup_n\vert x_n^*(x)\vert$ for all $x\in X$ and all $T\in\mathcal{M}$. It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: $\mathcal{M}$ is equicompact iff $\mathcal{M}^*=\{T^*\colon T\in\mathcal{M}\}$ is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set $\mathcal{M}\subset\mathcal{K}(X,Y)$ is equicompact iff each bounded sequence $(x_n)_n$ in $X$ has a subsequence $(x_{k(n)})_n$ such that $(Tx_{k(n)})_n$ is a converging sequence uniformly for $T\in\mathcal{M}$; 2) if $Y$ does not have finite cotype and $\mathcal{M}\subset\mathcal{K}(X,Y)$ is a maximal equicompact set, then, given $\varepsilon>0$ and a finite set $\{x_1,\ldots,x_n\}$ in $X$, there is an operator $S\in\mathcal{M}$ such that $\Vert Tx_i\Vert\leq(1+\varepsilon)\Vert Sx_i\Vert$ for $i=1, \ldots,n$ and all $T\in\mathcal{M}$.