\(q\)-deformed circular operators (Q2494150)

Let \(q\neq 1\) be a fixed positive real number. A closed densely defined operator \(T\) on a Hilbert space is said to have the property \(Q\), if \(T\) is unitarily equivalent to \(qT\). An example of an operator with this property is a \(q\)-normal operator, that is, an operator \(T\) such that \(TT^*=qT^*T\). The author shows that each nontrivial operator with the property \(Q\) is unbounded, and its spectrum contains zero. A bilateral weighted shift \(S\) with property \(Q\) is unitarily equivalent to \(qe^{i\theta }S\) for all real numbers \(\theta\). Such operators are called \(q\)-circular. It is shown that the spectrum of a \(q\)-circular weighted shift coincides with the whole complex plane. The location of its point spectrum, continuous spectrum and residual spectrum is investigated.