Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \cong H^0(j^{-1}\mathcal{N}_{X|\mathbb{P}^n})$?

EDIT Assume $Z$ is an irreducible subscheme of $X$.

Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \cong H^0(j^{-1}\mathcal{N}_{X|\mathbb{P}^n})$?

EDIT Assume $Z$ is an irreducible component of $X$.

Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Under what conditions can we conclude that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \cong H^0(j^{-1}\mathcal{N}_{X|\mathbb{P}^n})$?

Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \cong H^0(j^{-1}\mathcal{N}_{X|\mathbb{P}^n})$?

EDIT Assume $Z$ is an irreducible subscheme of $X$.