NCT Controller failed to comply with NCT/ZOA LOA requirements when descending an aircraft with Pitot tube icing issues without required coordination with ZOA.

Narrative

Aircraft X checked on frequency level at 12;000 MSL. I had another aircraft 10 miles in front of him opposite direction at 11;000 MSL. Aircraft X said that he was picking up light rime icing and needed lower. I informed him to expect lower shortly and issued traffic on the aircraft 1;000 FT below him. When the targets passed; Aircraft X said that his Pitot tube was freezing over and he needed lower right away. He stated that he lost his airspeed indicator. I immediately descended the aircraft concerned of his safety. Because I descended the aircraft without control from Center in Center's airspace; this event occurred. We need to standardize the LOA between NCT and ZOA to clarify who and when you have control of airplanes. Because [of where] this plane was landing; we do not have control of that airplane. If he was landing at SAC; my actions would have been praised; because we would have control for that airplane; and he continued on problem free. I think the LOA should say we have control for ALL aircraft on contact. Many other controllers express this concern as well. In an environment where safety is the main concern; should the aircraft have been told to wait to descend with failing instruments; while I got on the line with Center to coordinate? If I told the aircraft to wait for his descent; and got on the line to coordinate with Center; and his plane failed and crashed; would I then have to explain myself for coordinating control and not immediately descending him?