Implicit Operator Theorems under Group Symmetry Conditions

A G-invariant implicit operator theorem in stationary and nonstationary problems is derived on the basis of the general symmetry inheritance theorem for BEs and BERs without assuming that the allowed continuous group G is compact. In real Banach spaces, the general stationary branching problem is considered assuming that the nonlinear equation can be linearized in a neighborhood of the branch point. The operator intertwines the projectors with matrices with ones on the secondary subdiagonal and with zeros outside it. It is also found that the linear operator function has a tricanonical GJS. It is also shown that if there exists a complete tricanonical GJS, the problem of finding small solutions in is equivalent to finding small solutions of Lyapunov BER and Schmidt BER. The branch point moves along the trajectory and the relations that hold for linearization are found.