Let $H$ be a separable Hilbert space and let $\{e_i\}$ be an orthonormal basis. My first question is:

Is there a probability measure on $B(H)$ such that for $T$ chosen uniformly randomly the matrix entries $\langle Te_i, e_j \rangle$ are i.i.d. Gaussians?

In case I didn't formulate that quite correctly, I'm trying to imitate in infinite dimensions the familiar idea that that one can choose a "random matrix" by viewing its entries as i.i.d. Gaussians.

In case a probability measure is too optimistic, is there some alternative notion of "random bounded linear map" on Hilbert space that one can use? Also, if it is too much to hope that all matrix entries are i.i.d. Gaussian, it might be enough for this to be true "approximately" in some sense, though I won't try to formulate what that would mean until I get a better sense of what can go wrong.

Edit: As pointed out in the comments, my requirements that $T$ be bounded and that the matrix elements are i.i.d. are inconsistent. So I would like to eliminate the requirement that that the distributions of the matrix elements are identical, though I still want them to be independent. Most importantly, I am hoping for an actual measure on $B(H)$ (or a proof that no such measure exists).

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