For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology).
An array in the Banach space $X$ is a sequence of sequences, $(x_n^i)_{n=1}^\infty$, $i=1,2,\ldots$. We say the array $(x_n^i)_{n=1}^\infty$, $i=1,2,\ldots$, is weakly null if for each $i\in\mathbb{N}$, $(x_n^i)_{n=1}^\infty$ is weakly null. A diagonal of the array is a sequence of the form $(x^i_{n_i})_{i=1}^\infty$ for some positive integers $n_1<n_2<\ldots$.
Let $\mathcal{F}$ denote the set of non-empty, finite, strictly increasing sequences of natural numbers. That is, $$\mathcal{F}=\{(n_k)_{k=1}^i:k,n_1,\ldots,n_k\in\mathbb{N},n_1<\ldots<n_k\}.$$ Let $\varnothing$ denote the empty sequence. For $t\in \mathcal{F}\cup \{\varnothing\}$ and $n\in\mathbb{N}$, let $t<n$ denote the relation that either $t=\varnothing$ or $t=(n_k)_{k=1}^i\in \mathcal{F}$ and $n_i<n$. For $t\in\mathcal{F}\cup \{\varnothing\}$ and $n\in\mathbb{N}$ with $t<n$, we let $t\smallfrown n$ denote the concatenation. So $t\smallfrown n=(n)\in\mathcal{F}$ if $t=\varnothing$ and if $t=(n_k)_{k=1}^i$, $t\smallfrown n= (n_1,\ldots, n_i,n)$. We say a collection $(x_t)_{t\in\mathcal{F}}\subset X$ is weakly null if for each $t\in\mathcal{F}\cup \{\varnothing\}$, $(x_{t\smallfrown n})_{t<n}$ is a weakly null sequence. A branch of the tree $(x_t)_{t\in\mathcal{F}}$ is a sequence of the form $(x_{(n_1,n_2,\ldots, n_k)})_{k=1}^\infty$.
I'm going to skip the general definition of weakly null trees, which can be somewhat notationally tedious in the case that $X^*$ is non-separable. So there may be properties of interest which are characterized by weakly null trees for which it is not sufficient to check weakly null trees of the particular form mentioned in the previous paragraph. But that's an unnecessary detail for the scope of this question.
Lots of interesting/natural properties of Banach spaces or operators therebetween are determined by collections of sequences (bounded sequences, basic sequences, weakly null sequences, etc.). The characterization of the property typically has the form that every sequence from a particular class (bounded, basic, weakly null) has a subsequence with another property, or has a subsequence whose image under an operator has another property. Reflexivity, the Dunford-Pettis property, Banach-Saks property, weak Banach-Saks property, strict singularity, $\ell_p$-strict singularity, are all examples of such things.
Lots of interesting/natural properties of Banach spaces or operators therebetween are determined by trees. Containment of $\ell_p$, strict singularity, being an Asplund space/operator, having the Radon-Nikodym property. Characterizations of such properties typically have the form of "every tree (of a certain type) has a branch (with a certain property)."
In between sequences and trees, we have arrays. Every weakly null sequence $(x_n)_{n=1}^\infty$ gives rise to a weakly null array $(x^i_n)_{n=1}^\infty$, $i=1,2,\ldots$, where $x^i_n=x_n$ for all $i=1,2,\ldots$. The diagonals of this array are just the subsequences of the original sequence. Also, a weakly null array $(x^i_n)_{n=1}^\infty$ gives rise to a weakly null tree by letting $x_{(n_1,\ldots, n_i)}=x^i_{n_i}$ for each $(n_1,\ldots,n_i)\in\mathcal{F}$. The branches of this tree are just the diagonals of the original array.
Therefore if $\mathcal{C}$ is a class of sequences in the Banach space $X$, we can define a property by saying every weakly null sequence has a subsequence in $\mathcal{C}$, every weakly null array has a diagonal in $\mathcal{C}$, and every weakly null tree in $X$ has a branch in $\mathcal{C}$. By the previous paragraph, each property is implied by the subsequent one. These can be distinct properties. Indeed, if $\mathcal{C}$ is the class of norm-null sequences in $X$, the first two properties are equivalent to the Schur property, but the tree property is equivalent to being finite dimensional (here we do really need the full generality of "weakly null trees", not simply those indexed by sequences). Another more nuanced example is to take $\mathcal{C}$ to be the collection of $p$-weakly summing sequences in $X$ for some $1\leqslant p<\infty$, OR the collection of sequences $(x_n)_{n=1}^\infty$ such that there exists $C>0$ such that for every $N\in\mathbb{N}$ and every $N\leqslant k_1<\ldots <k_N$, $(x_{k_j})_{j=1}^N$ is $p$-weakly summing with $p$-weakly summing norm not more than $C$. For the latter example, the sequential property then becomes a statement about spreading models, whereas the tree property has to do with the asymptotic structures $\{X\}_N$ of $X$. Argyros et al have undertaken an extensive study of such properties to show that they are truly distinct, and that the array property is strictly stronger than the corresponding sequential property and strictly weaker than the corresponding tree property.
My question is: Are there any natural/commonly studied operator ideals which can be characterized by a condition on arrays, but which do not admit a characterization in terms of sequences or trees?
One could cheat and use the example of Argyros et al to come up with some ideals. For example, operators $T:X\to Y$ such that for each weakly null array $(x^i_n)_{n=1}^\infty \subset B_X$, $i=1,2,\ldots$, there exists a diagonal $(z_i)_{i=1}^\infty=(x^i_{n_i})_{i=1}^\infty$ such that all sequences of the form $(Tz_{k_i})_{i=1}^N$, $N\leqslant k_1<\ldots k_N$, are uniformly $p$-weakly summing. Then the identity operators on some of their examples would have this array property, but not the corresponding tree property, and other other of their examples would not have this array property, but they would have the corresponding sequential property. But this is hollow, as these classes are of interest, as far as I can tell, only because of the work already done to obtain those examples.
