On the closedness of the sum of subspaces of the space $B(H,Y)$ consisting of operators whose kernels contain given subspaces of $H$

Let $H$ be a Hilbert space and $Y$ be a Banach space. Denote by $B(H,Y)$ the linear space of all continuous linear operators $A:H\to Y$ endowed with the standard operator norm. For a closed subspace $H_0$ of $H$ denote by $Z(H_0;H,Y)$ the set of all operators $A\in B(H,Y)$ such that $Ax=0$ for every $x\in H_0$. It is clear that $Z(H_0;H,Y)$ is a closed subspace of $B(H,Y)$.

Let $n$ be a natural number, $n\geq 2$, and $H_1,\ldots ,H_n$ be closed subspaces of $H$. We will show that the following statements are equivalent: (1) $Z(H_1;H,Y)+\cdots +Z(H_n;H,Y)$ is closed in $B(H,Y)$; (2) $Z(H_1;H,Y)+\cdots +Z(H_n;H,Y)=Z(H_1\cap \cdots \cap H_n;H,Y)$; (3) the subspace $H_1^\bot +\cdots +H_n^\bot $ is closed in $H$.