# Characterization of operator ranges

My question is motivated by the following little proposition:

Proposition. For a vector subspace $$V$$ of a Banach space $$(X, \|\cdot\|_X)$$ the following assertions are equivalent:

(i) There exists a Banach space $$Z$$ and a bounded linear operator $$T: Z \to X$$ with range $$V$$.

(ii) There exists a complete norm $$\|\cdot\|_V$$ on $$V$$ such that the canonical embedding of $$(V, \|\cdot\|_V)$$ into $$(X,\|\cdot\|_X)$$ is continuous.

(See below for a proof.)

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Question. Are (i) and (ii) also equivalent to the following assertion (iii)?

(iii) There exists a bounded linear operator $$S: X \to X$$ with range $$V$$.

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Proof of the Proposition. Obviously (ii) implies (i), so assume that (i) holds. Let $$\tilde T: Z / \ker T \to X$$ denote the injective operator induced by $$T$$; then $$\tilde T$$ also has range $$V$$. The inverse $${\tilde T}^{-1}$$ is a closed linear operator $$X \supseteq V \to Z / \ker T$$, so $$V$$ becomes a Banach space with respect to the graph norm given by $$\|x\|_V := \|x\|_X + \|{\tilde T}^{-1}x\|_{Z / \ker T}$$ for all $$x \in V$$.

Remark. For Hilbert spaces results of this type can, for instance, be found in the paper "Fillmore and Williams: On Operator Ranges (1971)". In fact, the above proof is an adaptation of an argument that appears in the proof of Theorem 1.1 of this paper.

The answer to your question is "No". It can be seen in the following way: If there exists an operator $$S$$ mentioned in the Question, then, using the standard techniques, one can show that $$V$$ has to be isomorphic to a quotient space of $$X$$. So it remains to show that there exists $$X$$ and an operator range in $$X$$ for which this condition fails. This can be done by using injective nuclear operators with non-closed range from any separable Banach space $$V$$ into $$X$$ and by picking $$V$$ and $$X$$ in such a way that $$V$$ is not a quotient of $$X$$. For example, let $$X$$ be a separable Hilbert space and $$V$$ be a separable Banach space which is not isomorphic to a Hilbert space.