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We present adaptive optics‐assisted  K ‐band integral field spectroscopy of the central cluster galaxy in 2A 0335+096 ( z = 0.0349). The H 2  v=1–0 S(1) emission is concentrated in two peaks within 600 pc of the nucleus and fainter but kinematically active emission extends towards the nucleus. The H 2  is in a rotating structure which aligns with, and appears to have been accreted from, a stream of Hα emission extending over 14 kpc towards a companion galaxy. The projected rotation axis aligns with the 5 GHz radio lobes.  This H 2  traces the known 1.2 × 10 9  M ⊙  CO‐emitting reservoir; limits on the Brγ emission confirm that the H 2  emission is not excited by star formation, which occurs at a rate of less than 1 M ⊙  yr −1  in this gas. If its accretion on to the black hole can be regulated whilst star formation remains suppressed, the reservoir could last for at least 1 Gyr; the simultaneous accretion of just ∼5 per cent of the gas could drive a series of active galactic nucleus (AGN) outbursts which offset X‐ray cooling in the cluster core for the full ∼1 Gyr. Alternatively, if the regulation is ineffective and the bulk of the H 2  accretes within a few orbital periods (25–100 Myr), the resulting 10 62  erg outburst would be among the most powerful cluster AGN outbursts known. In either case, these observations further support cold feedback scenarios for AGN heating.

Molecular accretion in the core of the galaxy cluster 2A 0335+096