Convolution operators in the Fréchet sequence space $\omega = \pmb {\mathbb {C}}^{\pmb {\mathbb {N}}}$

Abstract

Each element $b = ( b_n)^\infty _{ n = 0 }$ of $\mathbb {C}^{\mathbb {N}},$ with $\mathbb {N}= \{ 0, 1, 2, \ldots \},$ convolves the discrete Fréchet space $\omega = \mathbb {C}^{\mathbb {N}}$ continuously into itself; denote this linear operator $x \longmapsto b *x$ by $T_b.$ Various properties of such operators are determined. For instance, the spectrum of $T_b$ is the singleton set $\{ b _0 \}.$ Furthermore, $b_0$ can either be an eigenvalue of $T_b$ or lie in the residual spectrum of $T_b,$ but never in the continuous spectrum. Every operator $T_b \ne 0$ is non-compact. Moreover, $T_b$ fails to be supercyclic for all $b \in \mathbb {C}^{\mathbb {N}}.$ It is shown that $T_b$ is not mean ergodic if b satisfies $| b_0 | > 1$ and also whenever $| b_0 | = 1$ with $b_0$ lying in the residual spectrum of $T_b.$ On the positive side, if b satisfies either $b_0 = 0$ or $\sum ^\infty _{ n = 0} | b _n| \le 1,$ then $T_b$ is mean ergodic.