Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another decomposition $\mathcal{X}=\mathcal{U}\oplus\mathcal{V}$ (of infinite-dimensional closed subspaces $\mathcal{U}$ with $\mathcal{V}$) with a continuous linear map $T\colon \mathcal{U}\to\mathcal{V}$ whose kernel in $\mathcal{U}$ or (closed) image in $\mathcal{V}$ is complemented?

If any partial/conditional results or known reformulations exist, that will be most appreciated.

Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another decomposition $\mathcal{X}=\mathcal{U}\oplus\mathcal{V}$ (of infinite-dimensional closed subspaces $\mathcal{U}$ with $\mathcal{V}$) with a continuous linear map $T\colon \mathcal{U}\to\mathcal{V}$ that admits a continuous right or left inverse?

If any partial/conditional results or known reformulations exist, that will be most appreciated.