I TA for a course in theory of computation and this came up as an interesting question.
$E_{TM}$ is the set of TM descriptions where the machine's language is empty.
Of course, $E_{TM}$ is undecidable for arbitrary TMs (as well as $A_{TM}$).
Now suppose we restrict our attention to decider TMs (ones that always halt).
$A_{decider}$ is the set of TM decider descriptions and strings such that the TM accepts that string, and $E_{decider}$ is the same as $E_{TM}$ except for deciders.
Of course, $A_{decider}$ now is decidable because we just simulate the decider on the input, and it will always stop.
Is $E_{decider}$ decidable now, or does it remain undecidable? The reduction to show that $E_{TM}$ was undecidable originally goes in the wrong direction here.