Let \(E_1\) with inner product \(\langle. \mid. \rangle_1\) and norm \(\|.\|_1\) and \(E_2\) with inner product \(\langle.\mid. \rangle_2\) and norm \(\|.\|_2\) be separable Hilbert spaces, and let \(H=E_1 \otimes E_2\) be the tensor product of \(E_1,E_2\) with inner product \(\langle . \mid .\rangle_H\), such that \(\langle h\mid h \rangle_H=\langle h_1\mid h_1 \rangle_1\langle h_2\mid h_2 \rangle_2\) for \(h=h_1\otimes h_2\). The domain, spectrum and spectral radius of a linear operator \(A\) in a Hilbert space are denoted, respectively, by \(Do(A)\), \(\sigma(A)\), \(r_s(A)\). The identity operators in \(E_1,E_2, H\) are denoted by \(I_1,I_2,I_H\), and the operator \(V\) is said to be a Volterra operator if it is compact and quasinilpotent, so that \(\sigma(V)= \{0\}\). In this paper, the author considers operators of the form \(A=D+(V_1^+ +V_1^-)\otimes I_2+I_1\otimes (V_2^++V_2^-)\), where \(V_1^+,V_1^-,V_2^+, V_2^-\) are Volterra operators satisfying \(P_j(t)V_j^+P_j(t)= V_j^+P_j (t)\), \(P_j(t)V_j^-P_j(t)= P_j(t)V_j^-\), \(j=1\) or 2, \(\overline P_1(t)\), \(\overline P_2(t)\) are resolutions of the identity, and \[ D=\int^\infty_{-\infty} \int^\infty_{-\infty} w(t,s)dP(t,s), \;P(t,s)=P_1(t)\otimes P_2(s),\;t,s\in \mathbb R. \] In the main results of the paper, the existence of the inverse operator \(A^{-1}\) and bounds for the operator norm are determined under the conditions \(d_0=\inf|\sigma(D) |>0\) and \(\psi_0J(V^\sim_1,V^\sim_2, d_0)<1\), where \(\psi_0\) and \(J(V^\sim_1,V^\sim_2,d_0)\) are defined in terms of \(V_j^+\), \(V_j^-\), \(j=1,2\). Special cases of the main result involve operators with Hilbert-Schmidt off diagonals and operators with Neumann-Schatten off diagonals.