Hypercyclic subspaces in omega
A continuous linear operator \(T\) acting on a Fréchet space \(X\) is called hypercyclic if there exists a vector \(x\in X\) whose orbit Orb\((T,x)=\{x, T(x), T^2(x), \ldots\}\) is dense in \(X\). Such a vector \(x\) is called a hypercyclic vector for \(T\). A hypercyclic manifold for \(T\) is a dense, invariant subspace of \(X\) consisting entirely -- except for the origin -- of hypercyclic vectors for \(T\). A hypercyclic subspace for \(T\) is a closed, infinite-dimensional subspace of \(T\) consisting entirely -- except for the origin -- of hypercyclic vectors for \(T\). It is well-known that once an operator on a Fréchet space has a hypercyclic vector, the smallest manifold invariant for \(T\) containing that vector is a hypercyclic manifold [see \textit{P.~Bourdon}, Proc.\ Am. Math. Soc. 118, 845--847 (1993; Zbl 0809.47005); \textit{D.~A. Herrero}, J. Funct. Anal.\ 99, 179--190 (1991; Zbl 0758.47016); \textit{J.~Wengenroth}, Proc. Am. Math.\ Soc. 131, 1759--1761 (2003; Zbl 1039.47009)]. The situation for hypercyclic subspaces is different. Indeed, \textit{A.~Montes--Rodríguez} [Mich.\ Math. J. 43, 419--436 (1996; Zbl 0907.47023)] showed that no scalar multiple \(\lambda B\), \(| \lambda | >1\), of the backward shift \(B(x_1,x_2,x_3, \ldots)=(x_2,x_3,x_4, \ldots)\) on \(\ell^2\) has a hypercyclic subspace. In particular, \textit{F.~Léon-Saavedra} and \textit{A.~Montes-Rodríguez} [J.\ Funct. Anal. 148, 524--545 (1997; Zbl 0999.47009)] showed that every separable, infinite-dimensional Banach space supports an operator with a hypercyclic subspace. This result holds also in the setting of Fréchet spaces with a continuous norm, as \textit{L.~Bernal-Gonzáles} [Proc. Am. Math. Soc. 134, No.~7, 1955--1961 (2006; Zbl 1094.47010)] and, independently, \textit{H.~Petersson} [J. Math. Anal. Appl. 319, No.~2, 764--782 (2006; Zbl 1101.47006)] proved. In the paper under review, the authors show that the space \(\omega\), i.e., the countably infinite product of the real or complex scalar field, endowed with the product topology, supports operators with a hypercyclic subspace, too, even though \(\omega\) is known not to have dense subspaces with a continuous norm.
scientific article; zbMATH DE number 5016164