Is the ātman 'me'?

Can one identify the absolute consciousness of Vedānta (the ātman) as "himself'"/"herself"? The problem lies in the fact that the ātman is beyond this-worldly subjectivity, whereas "I", "my", etc. are this-worldy concepts, at least according to Advaita Vedānta.
Wolfgang Fasching (in his contribution to the Sussex conference I already discussed) suggests that nonetheless the ātman is what "I" truly am. Independently of all this-wordly connotations, "I" am first and foremost an ātman. This seems to be linked with Husserl's (and Zahavi's) claims about the mineness of experience. So re-phrased, the question sounds: Can ther be an "I" (and a "my") beyond or before this-worldly subjectivity? If one transcends not only one's identification with the body and some similar accidents (e.g., one's first or family name), but also whatever belongs to one's being different from the others, is there still an I left?
Fasching answers affirmatively:

Yet one could reply that my present experience is mine (the experience I am experiencing) totally independent of any distinction I draw to what is not me (i.e. of my having an I-concept).

That is, my experience would be felt as "mine" even if I would not feel my experience and myself distinct from the others and their experiences. Is it really so? Can there be an I which does not ipso facto posit a non-I? Does not "my experience" presuppose that I am experiencing something different from the experiencer?
I am not asking a metaphysical question, but a phenomenological one. I am not interested in knowing whether experiencer and experienced are ultimately real, but whether the event of experience can be experienced as belonging to one without implying its postulating an experienced object.

Fasching quotes Zahavi saying that

first-personal givenness ‘is not a contrastive phenomenon’, it ‘does not arise thanks to any discrimination between self and the world […].

But, he then asks, why should one call this non-contrasted experience of subjecthood "mine"? Because, Fasching explains, eventually the non-conventional subject we identify which does not exist, and "we" are nothing but that non-contrastive experiencer. Hence, the ātman is ontologically 'me'. What about its being 'me' from a phenomenological point of view? I tend to doubt it. Do Vedāntic readers have other clues?