I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathcal{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I see a question of $L^2$ boundedness of an operator and some reference to Fourier coefficients, the use of the Plancherel theorem comes to mind. I think I know the answer to the following question, but I just wanted to check:
My question is, does there exist a Plancherel-type identity $$ ||f||_{\mathcal{H}^2}=||\hat{f}||_{\mathcal{H}^2} $$ for $f$ and $\hat{f}$ defined on the Hardy space on the unit circle $$ \mathcal{H}^2=\{f\in L^p(\mathbb{T}):\hat{f}(-n)=0~\forall~n\geq 1\}? $$ Why? If you wanted to prove or disprove it, would it be enough to examine the possibility of an isometry between $f\in\mathcal{H}^1(\mathcal{T})\cap \mathcal{H}^2(\mathcal{T})$ and $\hat{f}\in\mathcal{H}^2(\mathcal{T})$ across the Hardy space on the unit circle?
Would any of this hold on the $n-$dimensional torus $\mathbb{T}^n,~n\geq 2$
Thank you in advance for your help and comments.