Prescribed compressions of dual hypercyclic operators (Q2845738)

Let \(H\) be a separable, infinite-dimensional Hilbert space. A bounded linear operator \(T\in L(H)\) is said to be \textit{dual hypercyclic} if the operator \(T\) and its adjoint \(T^*\) are hypercyclic. The first example of such an operator was provided by \textit{H. N. Salas} in [Proc. Am. Math. Soc. 112, No. 3, 765--770 (1991; Zbl 0748.47023)]. Such operators can also be found in non-Hilbert spaces. \textit{H. N. Salas} proved that any Banach space with a separable dual space \(X^*\) admits a dual hypercyclic operator [Trans. Am. Math. Soc. 347, No. 3, 993--1004 (1995; Zbl 0822.47030)]. NEWLINENEWLINENEWLINEThe main result in the paper under review deals with the compression of dual hypercyclic operators in the Hilbert setting. In particular, it is shown how to construct a dual hypercyclic operator whose compression to a closed subspace of infinite codimension is prescribed. This result is stated as follows. NEWLINENEWLINENEWLINELet \(H\) be a separable, infinite-dimensional Hilbert space and let \(M\) be a closed subspace of \(\mathcal{H}\) with \(\dim(H/M)=\infty\). Let \(P:H\rightarrow H\) be the orthogonal projection onto \(M\). If \(A:M\rightarrow M\) is a bounded linear operator, then there is an operator \(T:H\rightarrow H\) such that: (i) \(T\) is dual hypercyclic, (ii) \(PTP|_M=A\), and (iii) \(PT^*P|_M=A^*\). In its proof, it is even constructed a common hypercyclic vector for \(T\) and \(T^*\). NEWLINENEWLINENEWLINEThis result can be compared with similar results on the compression of chaotic operators which were given by the author and \textit{G. Turcu} [Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., RACSAM 105, No. 2, 415--421 (2011; Zbl 1266.47014)] and by \textit{S. Grivaux} [Math. Z. 249, No. 1, 85--96 (2005; Zbl 1068.47011)].