Using suitably defined continuous analogs of the matricial circular systems and the direct integral of Hilbert spaces H " ş ' Γ Hpγqdγ, we study the operators living in H which give the asymptotic joint *-distributions of complex independent Gaussian random matrices with not necessarily equal variances of the entries. These operators are decomposed in terms of continuous circular systems tζpx, y; uq : x, y P r0, 1s, u P Uu acting between the fibers of H, the continuous analogs of matricial circular systems obtained when the Gaussian entries are block-identically distributed. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of the formula for the moments of T˚T , where T is a triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.2010 Mathematics Subject Classification: 46L54, 60B20, 47C15