Unfortunately, you might have trouble with light gun games. Some of the HDMI CRTs do some video processing that adds video delay. :(
If you're US based, you would probably use component video (YPbPr) to connect to your 1080i display; you can hook a ps2 light gun to the Y cable (usually green) and it should work if everything else is cool.
If you're US based, you would probably use component video (YPbPr) to connect to your 1080i display; you can hook a ps2 light gun to the Y cable (usually green) and it should work if everything else is cool.