2-regularity and reversibility of quadratic mappings (Q2704201)

This article deals with the problem about the boundedness of the right inverse \(Q^r\) for a continuous quadratic (homogeneous polynomial of degree 2) mapping \(Q\) between Banach spaces \(X\) and \(Y\). The author presents two examples of \(Q\) for which the operator \(Q^r\) is unbounded and proves the boundedness of \(Q^r\) provided that one of the following conditions holds:NEWLINENEWLINENEWLINE(1) \(N(Q)=\{0\}\), \(Q(X)= Y\) and there exists a number \(\alpha> 0\) such that NEWLINE\[NEWLINE\sup_{h\in N_\alpha(Q),\|h\|=1}\|Q(h,\cdot)^{-1}\|< \infty\qquad (N_\alpha(Q)= \{x\in X:\|Q(x)\|\leq\alpha\})NEWLINE\]NEWLINE andNEWLINENEWLINENEWLINE(2) there is \(h\in N(Q)\) such that \(\text{Im }Q(h,\cdot)= Y\); in both conditions \(N(Q)= \{x\in X: Q(x)= 0\}\) and \(Q(\cdot,\cdot)\) is the corresponding bilinear mapping between \(X\oplus X\) and \(Y\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].