Norm of right shift on finite dimensional space

Question

Let $X$ be a Banach space with a (normalized) Schauder basis $(e_{j})_{j=1}^{\infty}$. Then, for all $x\in X$, there exists a unique sequence of scalars $(\lambda_{j})_{j=1}^{\infty}$ such that \begin{equation} \limsup_{N\to\infty}\left\Vert x-\sum_{j=1}^{N}\lambda_{j}e_{j}\right\Vert =0 \end{equation} and the projection operators $P_{N}(x)=P_{N}\left(\sum_{j=1}^{\infty}\lambda_{j}e_{j}\right)=\sum_{j=1}^{N}\lambda_{j}e_{j}$ are known to be linear and (as $X$ is Banach) continuous. Consider the linear function $T:P_{N}(X)\to P_{N+M}(X)$ defined by: \begin{equation} T(x)=T\left(\sum_{j=1}^{N}\lambda_{j}e_{j}\right)=(0,\ldots,0,\lambda_{1}e_{1+M},\ldots,\lambda_{N}e_{N+M}) \end{equation} This function is automatically continuous because $\dim(P_{N}(X))<\infty$ and clearly, \begin{equation} \|T(e_{1})\|=\|e_{1+M}\|=1 \implies 1\leq\sup\{\|T(x)\| \mid x\in P_{N}(X) \text{ and }\|x\|=1\}=\|T\|_{\text{op}}.\end{equation} My intuition says that, in fact, the operator norm of $T$ should be equal to $1$ because this function would be an isometry if $X=\ell_{p}$ for any $p\in[1,\infty]$. Is this true? If so, could someone give me a hint as to how to prove it or suggest a reference? I have tried supposing $\|x\|<\|T(x)\|$ for some unit norm $x\in P_{N}(X)$ to derive a contradiction, but playing around with the inequality $0<\|T(x)\|-\|x\|$ hasn't yielded anything. I have also tried to use the fact that all norms are equivalent on a finite-dimensional space and this hasn't seemed to help either. Any suggestions would be greatly appreciated!

Answer 1

You meant $T(\sum\limits_{j=1}^N \lambda_j e_j)=\sum\limits_{j=1}^N \lambda_j e_{j+N}$, right? (You wrote the output in coordinates). However, this need not be an isometry, for example let $X=\mathbb{C}^3$, $e_1=(1, 0, 0)$, $e_2=(0,1,0)$ and $e_3=(0, \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$. For $N=2$, $M=1$ we have $||T(e_1+e_2)||=||e_2+e_3||\neq ||e_1+e_2||$ so $T$ is not an isometry. One case in which $T$ would be an isometry is if $X$ were a Hilbert space and $\{e_j\}_j$ were an orthonormal sequence.