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**ejgmath** I need to show that if $\displaystyle T:\ell^2(\mathbb{N},\mathbb{C})\rightarrow\ell^2(\ mathbb{N},\mathbb{C})$ is an operator defined by $\displaystyle (Tx)(j)=\frac{1}{2^j}x(j)$ and $\displaystyle \lambda\neq0$ is NOT in the set of eigenvalues of $\displaystyle T$ then $\displaystyle T-\lambda I$ is invertible. I also need to find the set of approximate eigenvalues of $\displaystyle T$ and the spectrum of $\displaystyle T$.

So far I have argued that since $\displaystyle \lambda$ is not a eigenvalue of $\displaystyle T$ and is not zero, there is no vector $\displaystyle x$ such that $\displaystyle Tx=\lambda x$ which implies that $\displaystyle Tx-\lambda x\neq0$ and therefore $\displaystyle Tx-\lambda Ix=(T-\lambda I)x\neq0$ and therefore $\displaystyle T-\lambda I\neq0$ hence $\displaystyle T-\lambda I$ is non singular and therefore invertible. Is this correct?