I don't know on that mower. I had a problem several years ago with a mower of mine. I found a service manual on ebay to load on my computer. About $10 to buy. It had all kinds of diagnostic test. I started at the battery, to the starter solenoid, the starter, fuses, safety switches, and so on. It turned out to be a fuse on my mower that I would have never found without the manual.

you can start here,

http://www.grasshoppermower.com/support_manual.php

15 yrs old.... pull the new battery - have it load tested at a qualified shop. M/W, take apart all electrical connections on both hot and ground side of starter circuit - clean to bare metal with emory/sandpaper - include both batt terms and chassis connections. Just a small amount of invisible corrosion can reduce starter power. Also chk ignition switch spade connections for clean good contact.
Can't hurt.

I replaced the starter. It starts right up repeatedly but shuts off right away. I have read about an anti-diesel solenoid but need help finding that or other opinions.

starving for fuel ? Either fuel is not getting to carb or carb is stopped up with water/trash. Take fuel line loose at carb - crank engine - see if there is a good fuel flow. If not, find/fix. If so, pull carb, clean it thoroughly - run fine copper wire thru jets/ports - blow out w/compressed air - re-asm with new gaskets.
Good luck.

May be a problem with the mower deck ingaged. With the engine off, does the mower blade clutch work?

I tried the mower clutch with the engine not running (item above dated Apr 21)and got no sound from the clutch on my grasshopper mower. Try this: Take the cover off the mower deck to expose the mower belts. Try to start the engine and see if the belts move. If they do, the mower blade clutch is engaged.

Problem solved - turned out to be 1 of 2 wires in the console that had mouse chewed insulation causing a short to ground.

Thanks for all suggestions