  We study the operators in the large $N$ tensor models, focusing mostly on the
fermionic quantum mechanics with $O(N)^3$ symmetry which may be either global
or gauged. In the model with global symmetry we study the spectra of bilinear
operators, which are in either the symmetric traceless or the antisymmetric
representation of one of the $O(N)$ groups. In the symmetric traceless case,
the spectrum of scaling dimensions is the same as in the SYK model with real
fermions; it includes the $h=2$ zero-mode. For the operators anti-symmetric in
the two indices, the scaling dimensions are the same as in the additional
sector found in the complex tensor and SYK models; the lowest $h=0$ eigenvalue
corresponds to the conserved $O(N)$ charges. A class of singlet operators may
be constructed from contracted combinations of $m$ symmetric traceless or
antisymmetric two-particle operators. Their two-point functions receive
contributions from $m$ melonic ladders. Such multiple ladders are a new
phenomenon in the tensor model, which does not seem to be present in the SYK
model. The more typical $2k$-particle operators do not receive any ladder
corrections and have quantized large $N$ scaling dimensions $k/2$. We construct
pictorial representations of various singlet operators with low $k$. For larger
$k$ we use available techniques to count the operators and show that their
number grows as $2^k k!$. As a consequence, the theory has a Hagedorn phase
transition at the temperature which approaches zero in the large $N$ limit. We
also study the large $N$ spectrum of low-lying operators in the Gurau-Witten
model, which has $O(N)^6$ symmetry. We argue that it corresponds to one of the
generalized SYK models constructed by Gross and Rosenhaus. Our paper also
includes studies of the invariants in large $N$ tensor integrals with various
symmetries.
