Update 14/02/2015
I such shortcomings in a Logitech product somehow could not imagine, I have a 2nd attempt is started:
Mouse M557 again ordered. And lo and behold, everything works. The connection is now strong and stable. It was the Bluetooth software automatically installed on the notebook (Setpoint extension) so that you can now make the mouse settings for this device on Logitech's Setpoint on the screen. Previously, I had assumed that the Bluetooth function on my Acer Aspire V5-573G was wrong; I was wrong: the first ordered mouse was probably a Monday edition and broken! The processing and the battery cover bracket are much better at this ??? How is that possible? Probably due to different production runs in the preparation. So folks, if your device does not work or has shaky parts, it exchanges to and tries again.