First some solutions, then I'll explain.

map(($_ => 2*$_), @list)
map +($_ => 2*$_), @list
map { $_ => 2*$_ } @list
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In general terms, what's an EXPR and what's not (and why can't $_ => 2*$_ be one)?

You gotta keep in mind the entire list of arguments is also an expression (a list operator), so the expression is bounded by a comma. "=>" is a fancy form of the comma operator.

So why you have is basically a precedence issue. And when you have a precedence issue, you use parens.

map(($_ => 2*$_), @list)
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map is often used without parens, but that's a problem here since

map ($_ => 2*$_), @list   # XXX
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means

map(($_ => 2*$_), ()), @list   # XXX
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The usual trick is to use the unary + operator.

map +($_ => 2*$_), @list
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Omitting parens around function in Perl is frought with peril.