I think that a generalization is present in “Around the circular law

” by Charles Bordenave and Djalil Chafaı̈, where they just assume that is uniformly integrable for every .

And shouldn’t the Frobenius norm be greater than the operator norm? is not a relaxation of

*[I am bounding , not – T.]*

Oops, I had forgotten here to impose moment conditions on the various measures involved. In this particular application the measure and also the empirical measures in fact all have uniform compact support, so this problem is avoided. (More generally, due to the very slow divergence of the logarithm function, virtually any moment condition will be fine here, e.g., it will be enough to obtain an almost sure bound on the Frobenius norm .)

]]>This is important since in the next line you use Minkowski, probably as follows:

Here the integral that is inside is not uniformly bounded on , and actually yields an unbounded function on , so the overall integral should depend on .

Probably I am wrongly interpreting the proof, so what is the correct line of reasoning here?

]]>There is no unique choice here, but one can for instance (after choosing a non-principal ultrafilter) take an ultraproduct of the finitary noncommutative probability spaces, and take the ultralimit of the finitary states, to obtain a limiting noncommutative space (where the limit will be along the ultrafilter, at least).

]]>In the perturbed shift example you just take the vector state for , but I’m not sure what the motivation is for this.

]]>Yes, this is the key identity that makes the Girko Hermitization method work (provided one can prevent the from getting too close to zero, of course.

]]>I’m trying to understand the chief advantage of the log potential. Is the log potential favored here mainly because (which allows us to consider measures on )? Or are there other important benefits to using the log potential which I failed to discern?

]]>I didn’t mean to say it wasn’t rigorous. Maybe I should just say I didn’t follow the proof.

]]>I’m having trouble converting this part from post-rigorous to rigorous, which I guess a mathematician could easily do. Could someone spell out the details for me? I have an ugly ad-hoc proof but it doesn’t really follow the high-level reasoning in the quote, so I’m curious what was intended here.

]]>Thank you! I will try to check this.

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