Can you determinize an NFA in PSPACE?
The subset construction can be computed (and negated) in polynomial space. Remember that the output doesn't count as part of the space usage in bounded space computation.

Determinization can exponentially increase the number of states: the language of $0$&ndash;$1$strings such that the $k$th-from-last character is&nbsp;$0$ can be recognized by an NFA with something like $k+1$ states, but any DFA for that language has at least $2^k$&nbsp;states.

So determinizing doesn't give you a polynomial space algorithm to determine if an NFA accepts its input.  But you can simulate all computation paths of the NFA simultaneously: as you read the input, just keep track of the set of states the automaton could be in, and see if that set includes an accepting state once you've read the whole input.