The notation to express a datum filtered by arbitray number of one-dimensional filters?
...my question is, is there an elegant notation to express such operation?


If you know that these operations are to act on orthogonal dimensions, you can define their 1D version and then use indexing to express the order by which the function is applied to the multidimensional signal.

But if you are working with multidimensional signals over an orthogonal vector basis throughout, you can simply specify this in the beginning and use (for example) $\mathcal{F}(x)$ throughout, to denote a multidimensional discrete Fourier transform.

Hope this helps (?)