Disjoint mixing operators
\textit{K. Chan} and \textit{J. Shapiro} [Indiana Univ. Math. J. 40, No. 4, 1421--1449 (1991; Zbl 0771.47015)] showed that each (non-trivial) translation operator \(f(z)\overset{T_\lambda}{\mapsto} f(z+\lambda)\) acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. The authors in the paper under review extend this result in the following sense. They show the existence of \(d\)-universal functions of exponential type zero for arbitrary commuting finite tuples of pairwise distinct translation operators. This is a consequence of a more general result in their paper (Theorem 1.3) about finite \(k\)-tuples of operators induced by series of powers of a ``backward shift'' operator. By a ``backward shift'' operator, they mean an operator \(T\) acting on a topological vector space \(X\) such that there exists an increasing sequence \(\{ X_n\}_{n\in\mathbb{Z_+}}\) of linear subspaces of \(X\) such that their sum is dense in \(X\), \(X_0=\{0\}\), and \(T(X_{n+1})\) is a dense subspace of \(X_n\) for each \(n\in\mathbb{Z_+}\). They also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of \(C_0\)-semigroups. They also provide an easy proof of the result of \textit{H. N. Salas} [J. Math. Anal. Appl. 374, No. 1, 106--117 (2011; Zbl 1210.47024)] that every infinite-dimensional Banach space supports arbitrarily large tuples of dual \(d\)-hypercyclic operators, and they construct an example of a mixing operator \(T\) acting on a Hilbert space so that \((T,T^2)\) is not \(d\)-mixing.
scientific article; zbMATH DE number 6073991
scientific article