For Banach space operators \(A, B\in B(\mathcal{X})\) , (A, B) is an m-isometric (m-symmetric) pair for some positive integer m if \(\triangle ^m_{A,B}(I)=(L_AR_B-I)^m(I)=0\) (resp., \(\delta _{A,B}^m(I)=(L_A-R_B)^m(I)=0\) ),where \(L_A\) ( \(R_A\) ) is the operator of left (resp., right) multiplication by A. Let \(D=\triangle \) or \(\delta \) . It is well known that if \(D_{D_{A_1,A_2},D_{B_1,B_2}}^m(I)=0\) , for operators \(A_i,B_i \in B(\mathcal{X})\) ( \(1\le i\le 2\) ), then \(D_{D_{A_1,A_2},D_{B_1,B_2}}^t(I)=0\) for all integers \(t\ge m\) . The reverse implication fails, and it is of some interest to find conditions on operators \(A_i, B_i\) guaranteeing \(D_{D_{A_1,A_2},D_{B_1,B_2}}^m(I)=0\) implies \(D_{D_{A_1,A_2},D_{B_1,B_2}}^t(I)=0\) for all positive integers \(t< m\) . This paper considers such conditions guaranteeing \(D_{D_{A_1,A_2},D_{B_1,B_2}}^m(I)=0\) implies \(D_{D_{A_1,A_2},D_{B_1,B_2}}(I)=0\) , and thence the range-kernel orthogonality \(\Vert X\Vert \le k \Vert D_{D_{A_1,A_2},D_{B_1,B_2}}(S) - X\Vert \) for some scalar \(k>0\) , all \(X\in D^{-1}_{D_{A_1,A_2},D_{B_1,B_2}}(0)\) and \(S\in B(\mathcal{X})\) . Amongst other results, it is proved that if \(A_i\) commutes with \(B_i\) : (a) \(B_i\) are invertible and the operator \(L_{A_1B^{-1}_1}R_{A_2B^{-1}_2}\) is power bounded, then \(\delta ^m_{\triangle _{A_1,A_2},\triangle _{B_1,B_2}}(X)=0\) implies \(\delta _{\triangle _{A_1,A_2},\triangle _{B_1,B_2}}(X)=0\) , which in turn implies \(\Vert X\Vert \le k \Vert \delta _{\triangle _{A_1,A_2},\triangle _{B_1,B_2}}(S) - X\Vert \) ; (b) \(A_1, A^*_2\) are hyponormal and \(B_1, B_2\) are normal (Hilbert space operators), then \(\delta ^m_{\triangle _{A_1,A^*_2},\triangle _{B_1,B^*_2}}(X)=0\) implies \(\delta _{\triangle _{A_1,A^*_2},\triangle _{B_1,B^*_2}}(X)=0\) , which in turn implies \(\Vert X\Vert \le k\Vert \delta _{\triangle _{A_1,A^*_2},\triangle _{B_1,B^*_2}}(S) - X\Vert \) .
