If $x\in{\rm I\! R}^n$, then diagonal matrix $\mathop{\rm diag}(x)$ is a linear operator $\mathop{\rm diag}(x): {\rm I\! R}^n \to {\rm I\! R}^n$.
I am curious if there is some analogy for infinite dimensional space, like if $f\in C[0,1]$ then can we define somehow $\mathop{\rm diag}(f): C[0,1] \to C[0,1]$?