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Obviously - since the Olympus device(s) work OK when provided with AC power - something is wrong in the battery power provisioning system.
Try cleaning the terminals in the battery compartment, after removing the batteries, first by rubbing off with a fresh pencil eraser, then by wiping the terminals off with cotton swabs moistened in rubbing alcohol, then letting them dry off for 5+ minutes with the battery compartment door open, then try turning the recorder On again.
How long has the battery or batteries in the camera been in-use? If over a year, then you might possibly have a battery/batteries that are getting weak. Rechargeable batteries do get weaker with time, and eventually fail to deliver enough power to start up a device like a digital camera; they then need replacing. The replacement battery needs to be charged up for 6+ hours before its first use, owing to its having been on a shelf for a considerable time.