B737 Captain reported a failure of the electric stabilzer trim from both pilot control yokes.

Narrative

Hand flying the aircraft on departure and climb. Everything was normal including trim function in both directions on preflight tests and in climb through about FL200. Somewhere in the low 20000 feet altitudes I tried to input some nose down trim and noticed there was no response. Alerted the First Officer and transferred the controls to him to see if it worked on his yoke but he got no response as well. I took the aircraft back and continued to fly while he troubleshot to see if any switches were out of place; circuit breakers popped; or any faults appeared in a recall. We found nothing. To continue trouble shooting we decided to see if the autopilot could trim. I engaged the autopilot on channel A and it was able to trim as needed both nose up and nose down. We leveled off at cruise with the autopilot engaged and then turned it off to ensure the aircraft was trimmed well. It was trimmed to level flight and our manual nose down inputs still had no response. We checked for any non normals but did not find a checklist for trim other than ones where all trim functions were inoperative. We re-engaged the autopilot and decided to get Dispatch and Maintenance Control involved. The First Officer took the flight controls and ATC comm while I setup a phone patch with Dispatch. We went over the details and Maintenance Control just asked if our stab trim switches were both in NORMAL; which they were. Since we had a fully functioning trim system with autopilot; plenty of suitable alternates along the route; and had reviewed the trim non normals in case the system failed; we felt comfortable to continue to ZZZ1. We asked Dispatch to relay our plan to the Chief Pilot and they all concurred we were good to continue. We kept the autopilot engaged until fully configured and about 1000 ft AGL on final approach. After disconnecting I found that the manual nose down trim input worked again. We landed without incident and left the aircraft with the trim malfunction maintenance write ups.