Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an infinite-dimensional hilbert space ?
In finite dimensions this would be true, since a projection there has always norm one.
Best Answer
It is true when $R,P$ are orthogonal to each other (there is an ambiguity in terminology, as "orthogonal" could mean that $PR=0$, or that $P=P^*=P^2$, $R=R^*=R^2$).
If $PR=0$ is not assumed, then the answer is no: take $P=R=I$, $a=b=1$, then $\|aR+bP\|=2$.
Assuming $PR=0$, then $\|aR+bP\|=\max\{|a|,|b|\}$. Indeed, for any $\xi$ in the range of $R$ with $\|\xi\|=1$, we have $$ \|(aR+bP)\xi\|=\|aR\xi\|=\|a\xi\|=|a|; $$ similarly, $\|(aR+bP)\eta\|=|b|$ if $\eta$ is in the range of $P$ and $\|\eta\|=1$. So $\|aR+bP\|\geq\max\{|a|,|b|\}$.
For an arbitrary vector $\nu$ with $\|\nu\|=1$, we can write $\nu=\alpha\nu_1+\beta\nu_2+\gamma\nu_3$, for three unit vectors with $\nu_1$ in the range of $R$, $\nu_2$ in the range of $P$, and $\nu_3$ orthogonal to both the ranges of $R$ and $P$, and $\alpha,\beta,\gamma\geq0$, $\alpha^2+\beta^2+\gamma^2=1$ (see edit below for an explanation). Then $$ \|(aR+bP)\nu\|^2=\|a\alpha\nu_1+b\beta\nu_2\|^2=|a|^2\,\alpha^2 + |b|^2\,\beta^2\leq\max\{|a|^2,|b|^2\}, $$ so $\|(aR+bP)\nu\|=\sqrt{\alpha^2|a|^2+\beta^2|b|^2}\leq\max\{|a|,|b|\}$.
In conclusion, $\|aR+bP\|=\max\{|a|,|b|\}$.
Edit: below is a proof of the claim that, given three pairwise orthogonal subspaces $X$, $Y$, $Z$ of a Hilbert space $H$ that span the whole space, any unit vector $\nu\in H$ can be written as $$\nu=\alpha\nu_1+\beta\nu_2+\gamma\nu_3,\ \ \ \nu_1\in X,\ \nu_2\in Y, \ \nu_3\in Z,$$ with $\alpha,\beta,\gamma\geq0$ and $\alpha^2+\beta^2+\gamma^2=1$.
Let $\{x_j\}$, $\{y_j\}$, $\{z_j\}$ be orthonormal bases for $X$, $Y$, $Z$. Together, they form a basis for the whole $H$. So there exist coefficients such that $$ \nu=\sum_j a_jx_j + \sum_jb_jy_j+\sum_jc_jz_j. $$ As $\|\nu\|=1$, $\sum_j|a_j|^2+\sum_j|b_j|^2+\sum_j|c_j|^2=1$. Let $$ \alpha=(\sum_j |a_j|^2)^{1/2},\ \beta=(\sum_j |b_j|^2)^{1/2},\ \gamma=(\sum_j |c_j|^2)^{1/2},\ $$ and $$ \nu_1=\sum_j\frac{a_j}\alpha\,x_j,\ \nu_2=\sum_j\frac{b_j}\beta\,y_j,\ \nu_3=\sum_j\frac{c_j}\gamma\,z_j. $$ Then $\nu_1,\nu_2\nu_3$ are unit vectors, $\alpha^2+\beta^2+\gamma^2=1$, and $$ \nu=\alpha\nu_1+\beta\nu_2+\gamma\nu_3. $$