Boundedness of an operator on a sum of spaces

Question

I have an operator $A \colon X \to X$ that is bounded on $X_0 \subset X$ and bounded on another subspace $X_N \subset X$. Is then $$ A\colon X_0 + X_N \to X $$ bounded? Or in other words, does it hold that for any $x_0\in X_0$ and $x_N\in X_N$ that $$\|A(x_0+x_N)\|\leq C\|x_0+x_N\|,$$ for a constant $C$?

If it helps... $X$ is a Hilbert spaces, and $X_N$ finite-dimensional.

Answer 1

I think, I got an answer. I will be happy to discuss it or to give the answered mark to any other useful answer.

Let's assume that the spaces are disjoint. (This is readily assured, if $X_N$ is finite dimensional).

Then, let $P$ be the projector that maps onto $X_N$ along the orthogonal complement of $X_0$. Such a projector is bounded. Let's say, in this case, with a constant $C_P$.

Then, with $x = x_0 + x_N = [I-P]x + Px$, we can estimate

$$ \|Ax\| = \|A x_0 + Ax_N \| \leq \|A\|\|[I-P]x\| + \|A\|\|Px\| \leq C(C_P + 1 + C_P)\|x\| $$ which proves the boundedness on $A$ on $X_0+X_N$.