I'm trying to understand the spectrum and resolvent of a linear operator on a Banach space in as much generality as I possibly can.(adsbygoogle = window.adsbygoogle || []).push({});

It seems that the furthest the concept can be "pulled back" is to a linear operator [itex]T: D(T) \to X[/itex], where [itex]X[/itex] is a Banach space and [itex]D(T)\subseteq X[/itex]. But here are a few questions:

(1) Doesn't [itex]D(T)[/itex] necessarily have to be asubspaceof [itex]X[/itex] for this concept to make any sense? For instance, if it's not a subspace, there can be elements [itex]x,y[/itex] for which [itex]Tx[/itex] and [itex]Ty[/itex] make sense, but for which [itex]T(x+y)[/itex] makes no sense, since [itex]x+y[/itex] might not be an element of [itex]D(T)[/itex].

(2) The Wikipedia article here says that, in order for [itex]\lambda \in \rho(T)[/itex], we need both (1) [itex](T-\lambda)^{-1}[/itex] exists and (2) [itex](T-\lambda)^{-1}[/itex] is defined on a dense subset of [itex]X[/itex]. Is this equivalent to saying that [itex]\text{Ran} (T-\lambda)[/itex] must be a dense subset of [itex]X[/itex] and that [itex](T-\lambda): D(T) \to \text{Ran} (T - \lambda)[/itex] must be a bijection?

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# Spectrum of a linear operator on a Banach space

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