IanB writes

surely it will be operated by the (then) existing TOCs (as with Thameslink)
so not new TOC & nothing to do with LUL (apart from shared stations). Can
you use your LUL staff pass on Thameslink trains from Hendon to Croydon or
West Hampstead to Elephant & Castle?... I would imagine that the same rules
would apply.

Cross London Rail Links Ltd has been set up by TfL and the SRA. As to
who operates it, well that's so far into the future that it's almost not
worth bothering about.

My guess would be that if it ever gets built, it'll be a
design/build/operate lease a la Tramlink.

www.crossrail.co.uk

--
Dave