Generalized bicircular projections via the operator equation αX2AY2+βXAY+A=0

The operator equation αX2AY2+βXAY+A=0 is used to determine the structure of a projection P such that for some complex unit λ≠1 the mapping P+λ(I-P) is A↦UAV,A↦UA∗V, or A↦UAtV for some fixed operators U,V. Some applications of these results are also given.