Unbounded 2-hyperexpansive operators (Q2758151)

A linear operator \(T\) acting in a Hilbert space \({\mathcal H}\) isNEWLINENEWLINENEWLINE(i) 1-hyperexpansive if \(\|Tf\|\geq\|f\|\) for \(f\in{\mathcal D}(T)\);NEWLINENEWLINENEWLINE(ii) 1-isometric if \(\|Tf\|=\|f\|\) for \(f\in{\mathcal D}(T)\);NEWLINENEWLINENEWLINE(iii) 2-hyperexpansive if \(\|T^2f\|^2- 2\|Tf\|^2+\|f\|^2\) for \(f\in{\mathcal D}(T^2)\);NEWLINENEWLINENEWLINE(iv) 2-isometric if \(\|T^2 f\|^2- 2\|Tf\|^2+\|f\|^2\) for \(f\in{\mathcal D}(T^2)\).NEWLINENEWLINENEWLINEThe authors prove unbounded 2-hyperexpansive operators have most properties of (bounded) completely hyperexpansive operators and that powers of closed 2-hyperexpansive operators are again closed and 2-hyperexpansive. There are also examined parts of spectra of such operators, properties of 2-hyperexpansive weighted shifts and some closed 2-hyperexpansive (2-isometric) operators with invariant dense domains.