The datasheet of the LM4565 shows that it is not rail to rail on output or input.
Input range is 14V when operated on 15V, so one volt less that the rail voltage.
The output range is somewhere above 11V when operated on 15V. So, output range is a couple of volts less than rail voltage.
It says it's rated for operation from +-2V to +-18V. So, single sided 4V to 36V.
Your power supply is outside the specified limits, and even on 4V it would be of little use.
This is the relevant section of the datasheet:
Look at the lines marked "Maximum Output Voltage" and "Input Common-mode Voltage Range," and note that those voltages are for when the IC is operating on one 15V rail.