Open mapping theorem and direct sum

Question

Let $T$ be an invertible operator on a Banach space $X$. Suppose that $X=M \oplus N$ where $M$ and $N$ are closed subspaces of $X$, $T(M)\subseteq M$ and $T(N)\subseteq N$.

In a paper, author claim that by open mapping theorem there is $\beta>0$ such that $||a||<\beta ||x||$ and $||b||<\beta||x||$ where $x=a\oplus b$. Proof of it is not clear for me. I so thankful if give me a reference or idea about it.

Answer 1

$X=M\oplus N$ where $M$, $N$ are closed subspaces. Define $P:X\to M$ by $Px = a \Leftrightarrow x=a+b$. Then, $P$ is a bounded linear operator with the property $P^2 = P$.

\begin{eqnarray} \|a\| &=& \|Px\|\leq \|P\|\|x\| \\ \|b\| &=& \|(I-P)x\|\leq \|I-P\|\|x\| \end{eqnarray}

Let $\beta = \max\{\|P\|,\|I-P\|\}$.