A frequent reviewer seems to have a grudge against me. What to do? (Asking on behalf of a friend.)

I work in math, in a field where reviews are not double-blind: reviewers are anonymous, but review copies of papers include attribution.  I’ve received several reviews recently (on different submissions) which were similar in style, all negative, and rather light on substantive criticisms.  The other reviews for those submissions were generally positive.

I’m rather concerned that they’re from a single reviewer with a grudge against me — it’s a small field, so one can easily encounter the same reviewer/author repeatedly — and I have some suspicion of who this reviewer might be (though far from certain).

Is there any defensive action I can take on future submissions — anything I can say to the editor at some stage to avoid getting shot down by this same reviewer again?
I've had a similar experience, but on the other side of the table.  I was the reviewer (the journal was JAES, the Journal of the Audio Engineering Society).

There was an author who published a boat-load of papers on a subject where I have published just one.  But my paper dealt with some very fundamental issues (I called the paper "Wavetable Synthesis 101, A Fundamental Approach" or similar).  The AES Editor did not send any of this author's papers in my direction and when enough of these papers that made only incremental advancements did get published in the Journal, I finally wrote a Letter to the Editor (that was published) that was a little critical, because this author consistently ignored a mathematical issue (phase) that my 101 paper did not.

Anyway, after about the 10th or 15th new Wavetable paper to come from this author, the JAES editor did finally send a manuscript to me to review.  I was not nasty, but I did continue to point out what this author was obstinately choosing to side step (offering no mathematical justification).

I am sure the author knew who that reviewer was, even though I was supposed to be anonymous.

So, if the reviewer with a grudge is picking on you for just anything, maybe the reviewer just doesn't like you and there could be any number of bad reasons.  Maybe he/she thinks you're encroaching on his/her turf and/or hadn't paid your dues yet.  Maybe this reviewer wants to be cited by you where he isn't.  Who knows?

But if this reviewer is picking on you for about the same reason, on these multiple submitted manuscripts, I might, if I were you, think about what that particular reason is and answer it one way or another.  One might be a consistent technical omission.  Or maybe, just like the "bad reviewer" above, this reviewer thinks you're missing critical citations and his/her work is among the missed citations.

Whether the reviewer has a grudge or not, he/she is not the only reviewer.  But you still are obligated to take seriously every criticism and suggestion for improvement offered by the reviewers.  It doesn't mean that you will agree with every one, but then be prepared to respond to each critique or suggestion that you disagree with.  Persuade the reviewer, the other co-reviewer (who is not picking on you) and the editor, that the critique is mistaken.

Finally, if, after all this, you feel that this reviewer is useless, find some way of asking the journal's editor or assistant editor (whoever has dispatched your submission to whoever appropriate reviewers) to consider getting a third or fourth opinion.  Even suggest a couple of persons in the field that you respect but make sure they are not in the same institution you are in nor have ever co-authored a paper with you.