Raster Split Tool creates NoData Frame for output clipped raster. Background values cannot be discarded on ArcMap when this nodata one pixel wide frame created . Also I could not see any nodata frame at base pyramid or data level.
If you do not see the 'frame' I would consider the resampling method of the pyramids. If you have bilinear, then I would recommend trying to create the pyramids with nearest neighbor. If you see the reference in this help document, you can see what this will fix. To modify the resampling method use the Build Pyramids tool and then modify the environment settings to change the resampling method. [ATTACH=CONFIG]11174[/ATTACH]
