On some properties of Banach operators (Q1599710)

Generalizing the notion of contractions, an operator \(\alpha : X\to X\) is called Banach operator if there is a constant \(0\leq k <1\) such that \(\|\alpha^2(x)-\alpha (x)\|\leq k \|\alpha (x)-x\|\) for all \(x\in X\), where \(X\) denotes a normed spaces endowed with norm \(\|\cdot\|\). The concept of Banach operators was introduced in [\textit{S. A. Naimpally, K. L. Singh} and \textit{J. H. M. Whitfield}, Fundam. Math. 120, 63-75 (1984; Zbl 0561.47047)], and it is of special interest in fixed point theory. The note under review is aimed at investigating some properties of Banach operators. Along these lines, some decomposition results are shown first. E.g., if \(\alpha\) is a linear Banach operator on normed space \(X\), then (i) \(N(\alpha -1)=N((\alpha -1)^2)\), (ii) \(N(\alpha -1)\cap R(\alpha -1)=\{0\}\), where respectively, \(N(\cdot)\) and \(R(\cdot)\) denote the null space and the range of some operator \(\cdot\), and \(1\) the identity mapping. Considering the special case that \(X={\mathcal H}\) is a Hilbert space, it is shown that the following are satisfied for a linear contraction \(\alpha\): (i) \(N(\alpha -1)\cap R(\alpha -1)=\{0\}\), and (ii) \(N(\alpha -1)+R(\alpha -1)\) is dense in \({\mathcal H}\). Using the techniques developed in the note under review, a new proof for the known fact is given that the residual spectrum of a bounded normal operator on a Hilbert space is empty. Assuming finally that \(X\) is a normed algebra with unity, some commutativity results are obtained for a pair of invertible bounded linear Banach operators \(\alpha , \beta\) satisfying the operator equation \(\alpha +c\alpha^{-1}=\beta +c\beta^{-1}\) on \(X\), where \(c\) is an appropriate real or complex number. Certain situations are identified such that the above operator equation implies, respectively, \(\alpha\beta =\beta\alpha\) and \(\alpha =\beta\).