i looked to the question Spectrum and point spectrum of this operator. I will go further with asking. We know that $T$ is well-defined iff $(\lambda_n)\in\ell^{\infty}$. But if $(\lambda_n)\notin\ell^{\infty}$ then we can defined this operator also on a new domain $D(T)$ given by $D(T)=\{\sum_{n=0}^{\infty}{x_ne_n}\in\ell^2: \sum_{n=0}^{\infty}{|\lambda_nx_n|^2}&lt;\infty\}$. My first question is: Is this $D(T)$ dense in $\ell^2$? 

I observe that $T$ is well-defined on $D(T)$. Thus we can also ask about the spectrum of $T$. We have seen in Spectrum and point spectrum of this operator that the spectrum is $\sigma(T)=\overline{\{\lambda_n:n\in\mathbb{N}\}}$. Can we say this also for $T$ defined on $D(T)$? Is it the same argumentation?

Thank you for help.


Question: " not all code paths return a value" when return enum type
Answer: Maybe you're sure that a return type will always be given, but the compiler isn't [imagine that all the if conditions fail - ie: a variable was changed while your code was executing by an external program.  Then what would happen?]

Just stick a return at the bottom as a "default" value.  You could also throw an exception too, if you want, since the bottom should never be reached, right?