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**ejgmath** I need find all the eigenvalues of the operator $\displaystyle T:\ell^2(\mathbb{N},\mathbb{C})\rightarrow \ell^2(\mathbb{N},\mathbb{C})$ defined by $\displaystyle (Tx)(j)=\frac{1}{2^j}x(j)$ for all $\displaystyle j$ and show that $\displaystyle \lambda =0$ is an approximate eigenvalue.

So far I have deduced that since $\displaystyle (Tx)(j)=\frac{1}{2^j}x(j)$ and $\displaystyle \lambda$ must satisfy $\displaystyle Tx=\lambda x$ for $\displaystyle x\in\ell^2$ with $\displaystyle x\neq 0$ that $\displaystyle \lambda_j=\frac{1}{2^j}$, is this correct?