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

Abstract

Each element $$ b = ( b_n)^\infty _{ n = 0 } $$ of $$ \mathbb {C}^{\mathbb {N}},$$ with $$ \mathbb {N}= \{ 0, 1, 2, \ldots \},$$ convolves the discrete Frechet 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.