In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $p$ the operator $\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(\mathbb{Z})\backslash \mathbb{H})$ is introduced to prove the existense of inifinitely many even cusp forms and consequently even maass forms.
Cant we just just take the collection of functions given by $h(y)\cos (2\pi jx)$ where $h(y)$ is real valued smooth function supported in $[1,2]$ and extend it to functions on $SL_2(\mathbb{Z})\backslash \mathbb{H}$. As $j$ varies through positive integers this already gives infinitely many independent even cuspidal functions. Why introduce $\heartsuit$ ?