HDMI interface includes a security / copy protection feature called 'HDCP'.
Some, but not all, DVI interfaces support that. I haven't been able to figure out if the L-2266WD has that or not. But if it doesn't then it will not work.
Here are some things to try.
With the PS3 running, and cable connected to your monitor, unplug your monitor power cable and then plug it back in and turn on monitor. Do you get any beeps or flashing when you plug it back in or turn it on?
If that doesn't work with your monitor powered on try to press the power button on the monitor and hold it for maybe 10 sec. Do you get any beeps or flashing then?