I have been studying functional analysis lately, specifically spectrum of operators. I know how to find the spectra of an operator, but what if I have the spectra and I want to find an operator with such spectra?
Let's say I have $\delta(T) = \{3i\}$ as the spectra of some linear and bounded operator T. I guess that T could be something like this:
$ T: l^2 \to l^2$, such that $\{x\}_j \to 3i\{x\}_j$ could be a trivial candidate because $Tx = \lambda x$. However what if I have $\delta(T) = \{3i,5i,7i\}$ for example?
It is simple to do so because the multiplication operator $T: L^2(\mathbb{R}) \to L^2(\mathbb{R})$, $f \mapsto g f$, with some bounded function $g: \mathbb{R} \to \mathbb{C}$, has as spectrum the closure of the range of $g$.
So for your first example, you take the constant function $g: g(x) = 3i$ and get the spectrum $\{ 3i\}$. Similarly you can choose any function with $g(\mathbb{R}) = \{3i,5i,7i\}$ for your second example (take something peacewise constant for example) or similarly in any example you can think of.