Let $H$ be a Hilbert space and let $\{e_n, n \geq 1\}$ be an orthonormal basis for $H$.
a) Determine the operator $T\in B(H)$ that satisfies $ Te_1 = 0,\; Te_n = \frac{1}{n}e_{n-1}, n >1.$ b) Show that $T$ is compact and determine its spectrum.
My try:
$Tx = T(\sum_i \alpha_i e_i) = \sum_i\alpha_iTe_i = \sum_{i=2}^\infty \frac{1}{i}\alpha_{i}e_{i-1},$ where $x = \sum_i \alpha_ie_i.$
Is this correct?
For b) Im not sure how to prove that the image of T is pre compact and how do I find the spectrum?