Define a pair of operators $S\colon\ell^2 \rightarrow \ell^2$ and $M\colon\ell^2 \to\ell^2$ as follows: $S(x_1,x_2,x_3,x_4,\ldots)=(0,x_1,x_2,x_3,\ldots) $ and $M(x_1,x_2,x_3,x_4,\ldots)=(x_1,\frac{x_2}{2},\frac{x_3}{3},\frac{x_4}{4},\ldots).$ Let $T=M \circ S$.
Then how to prove that $T$ has no eigenvalues? Is $T$ a compact operator? And self-adjoint? Also, if let $T^m$ denote the composition of $T$ with itself $m$ times. Then how to show directly from the formula for $T$ that $\lim_{m\rightarrow \infty} \|T^m\|^{1/m}=0$.
I think I need to use the Stirling's formula, but I don't know what to do. Please help me. Thank you.