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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?

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The easiest way to show that $T$ is compact is to show that it is a norm limit of finite-rank operators.

But it is not hard to show that the image is pre-compact if you look at the particular kind of sequences you get in the range of $T$.

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