Generalized backward shifts on Banach spaces. (Q5944905)

For a bounded linear operator \(T\) on a Banach space \(X\), \(\text{Ker\,}T\) and \(\text{Ran\,}T\) denote respectively the kernel and range of \(T\). \(T\) is said to be a generalized forward shift if (1) \(T\) is an isometry on \(X\), (2) \(\text{Ran\,}T\) is of \(\text{codim\,}1\), (3) \(\bigcap_{n\geq 1}\text{Ran\,}T^n= \{0\}\).NEWLINENEWLINE \(T\) is said to be a generalized backward shift if (1) \(\dim\text{Ker\,}T= 1\), (2) \(\widehat T\) induced by \(T\) on the quotient space \(X/\text{Ker\,}T\) is an isometry, (3) \(\bigcup_{n\geq 1}\text{Ker\,}T^n\) is dense in \(X\).NEWLINENEWLINE Is there a generalized forward and backward shift on every Banach space? This is an unsolved basic problem. In the paper under review, the authors discuss the problem of existence of generalized backward shifts in the case of the classical Banach spaces \(C(F)\) and \(L^p(\mu)\), \(1\leq p\leq\infty\). The basic properties of generalized backward shifts are also established.