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Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.

Can the same be done for a family of projections? That is, given finitely many projections $P_1$,...,$P_n$ in $B(X)$ is there an equivalent norm under which all projections become contractive?

Are there any assumptions other than $P_i$ commuting that would ensure this?

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    $\begingroup$ The necessary and sufficient condition is that the semigroup generated by the projections be uniformly bounded. As Andreas pointed out, this need not be true even in the plane. $\endgroup$
    – Bill Johnson
    Commented Jun 23, 2016 at 18:27

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Let $X$ be the Euclidean plane, and consider the two projections $$ P_1=\begin{pmatrix}1&A\\0&0\end{pmatrix}\quad\text{and}\quad P_2=\begin{pmatrix}0&0\\A&1\end{pmatrix}, $$ where $A$ is a big constant (actually any $A>1$ will do). If a norm on $X$ gave both of them norms $\leq1$, then the same would be true for their product, and for powers of their product,which have the form $$ (P_1P_2)^n=\begin{pmatrix}A^{2n}&A^{2n-1}\\0&0\end{pmatrix}. $$ But these powers send the vector $\binom10$ to larger and larger vectors, which cannot all be in any ball.

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