Can a Turing machine determine if a set is accepted by a another Turing machine?
Is the language of all machines that accept $W$ recognizable?

$$L_W = \{\langle M\rangle : L(M) = W\}$$

The answer is unfortunately no, except in trivial cases. If $W$ is unrecognizable, then $L_W$ is empty (no machine recognizes $W$) and hence decidable (The machine that rejects all inputs decides $L_W$).  In all other cases, $L_W$ is unrecognizable. We can show this using reductions from the unrecognizable rejection problem&mdash; the problem of determining whether a machine rejects a word.


If $W=\Sigma^*$, the set of all strings, then $L_W$ is unrecognizable (a standard result: "$M_x(w)$ simulates $M$ on $x$ for $|w|$ steps and accepts if the simulation doesn't accept by then." provides a reduction from the problem of deciding whether $M$ rejects $x$).
If $W\neq \Sigma^*$ is decidable, then $L_W$ is unrecognizable. Consider a program that takes a machine and word $\langle N, x\rangle$ and builds a new machine $N_x(w)$. The new machine decides whether $w\in W$ and accepts if so. Otherwise, it simulates $N$ on $x$ and does what it does. The machine $N_x$ therefore accepts $W$ (if $N$ rejects $x$) or all strings (if $N$ accepts $x$).

If we could recognize whether $N_x$ accepts $W$ or not, we could recognize whether $N$ rejects $x$ or not&mdash;an impossibility.
If $W$ is recognizable but undecidable, then $L_W$ is unrecognizable:  Consider a program that takes a machine and word $\langle N, x\rangle$ and builds a new machine $N_x(w)$. The new machine simulates $N$ on $x$ for $|w|$ steps. If the simulation accepts in that time, the machine accepts. Otherwise, the machine tests whether $w\in W$ and does what it does. 

Hence $N_x$ accepts the union of $W$ and the set of all strings longer than the number of steps it takes for $N$ to accept $x$. If $N$ rejects $x$, $N_x$ recognizes $W$. If $N$ accepts $x$, $N_x$ recognizes additional words. (Note that because $W$ is undecidable, there must be arbitrarily long strings not in $W$. Otherwise, a DFA could decide $W$ by memorizing the short strings and using a length requirement for the long ones. Hence $N_x$ recognizes a strict superset of $W$ if $N$ accepts $x$.)

If we could recognize whether $N_x$ accepts $W$ or not, we could recognize whether $N$ rejects $x$ or not&mdash;an impossibility.