Split exact sequence and reflexive tensor product.

Question

Let $X$ be a Banach space and $T \in \mathbb{B}(X)$ is Fredholm operator (I.e the range of $T$ is closed, the ker of $T$ has a finite dimension and the dimension of the co-kernel ($X/ R(T)$) of $T$ is finite ). Clearly, $$0 \rightarrow \ker(T) \rightarrow X \rightarrow T(X) \rightarrow 0 $$ is a split exact sequence. I have the following questions: \ 1- Let $H$ be a reflexive Banach space. Is the following sequence $$ 0 \rightarrow \ker(T) \widehat{\otimes } H \rightarrow X \widehat{\otimes } H \rightarrow T(X)\widehat{\otimes } H \rightarrow 0$$ an exact sequence ?. Where $X \widehat{\otimes }H$ denotes the topological tensor product completed with respect to the greatest crossnorm of Schatten.\ 2- $\ker(T) \widehat{\otimes } H$ is it reflexive ?. Please, I want explanatory answers to the previous questions. Thanks.