An air carrier Captain reported an NMAC on approach to OAK when his flight path conflicted with a light aircraft off HWD who was reportedly deviating from his clearance.

Narrative

As we were being switched to Tower frequency from OAK TRACON; I visually spotted a light single-engine aircraft at our 12 o'clock position and it was climbing. His estimated the range to us was approximately one to one and one half miles and his altitude was approximately 700' AGL. I would estimate our altitude at the time to be about 1400' AGL and descending. As I checked in with OAK Tower; I stated I had traffic ahead in sight and then noticed the Cessna was still climbing. The OAK Tower issued a Traffic Advisory and then cleared us to land. About the same time; the TCAS issued a 'Climb; Climb' command. The First Officer disconnected the autopilot and complied with the TA. My best guess is that we climbed about 200'-300' and missed the climbing Cessna by about 300' vertically. Laterally; I believe he was right below us. After the 'Clear of Conflict' was issued; we continued the visual approach to a normal landing. ATC stated that the Cessna had violated his altitude clearance climbing out of Hayward Airport and was being tracked down. I was told to call OAK Tower on arrival and did so at the gate. The Tower Chief asked my opinion as to how close we thought we had passed the Cessna. I relayed to him the same information as stated above. Maybe the Tower Controllers at Hayward and Oakland could better coordinate air traffic so departing Hayward traffic is timed to not interfere with arriving OAK traffic. This way; a conflict would not develop with a single failure; e.g.; a missed altitude clearance would not lead to a conflict or a near miss.