No this is not possible in normal javascript. You can't define custom operators, only objects and therefore functions. You also can't change javascripts syntax for calling functions.
If you really want to use syntax like this you can use coffeescript and write code like this
typeOf = (x) -> "whatever you want"
typeOf "string"
which will translate to javascript like this
var typeOf;
typeOf = function(x) {
return "whatever you want";
};
typeOf("string");
Example
But there's no way to replicate that in pure javascript.
A quick aside
One more note: I wouldn't advise naming this operator typeOf even if you did have it. Naming 2 things that act similarly but act slightly different the same thing other than a single capitalized letter is just begging for subtle typo errors that are hard to trackdown and check. Something like exactType
or something else different would remove the potential confusion/bugs.