When can we get discrete spectrum?

Question

Suppose that $T$ is a densely defined closed operator on a separable Hilbert space $H$. Form $N = T^*T$. Assume further that $T$ has a finite dimensional kernel and satisfies the commutation relation $$ TT^* - T^*T = 1 $$ Is it always true that $N$ has discrete countable spectrum (namely, $\sigma(N) = \{0,1,2,3,...\}$) and $N+1$ is the inverse of a compact bounded operator? (It seems this is true for most interesting cases in QM)