While digging through old piles of notes and jottings, I came across a question I'd looked at several years ago. While I was able to get partial answers, it seemed even then that the answer should be known and in the literature somewhere, but I never knew where to start looking. So I thought I'd ask here on MO if anyone knows of a reference for this observation.

Here's the notation and background for the question. Let $(E,\Vert\cdot\Vert)$ be a real, normed vector space (I think the complex case works out to be almost identical). We will see shortly that my question is vacuous unless $E$ is incomplete. Let $F$ be the completion of $E$.

Denote by $B(E,\Vert\cdot\Vert)$ the space of all linear maps $T:E\to E$ which ae bounded with respect to $\Vert\cdot\Vert$, i.e. there exists $C$ depending on $T$ such that $$\Vert T(x)\Vert \leq C\Vert x\Vert \;\;\hbox{for all x\in E.}$$

Clearly each $T\in B(E,\vert\cdot\Vert)$ extends uniquely to a bounded linear operator $F\to F$, and we thus get an injective algebra homomorphism $\imath:B(E,\Vert\cdot\Vert)\to B(F)$. The question arises: when does $\imath$ have dense range?

• It is not hard to show that if $E=c_{00}$ and $\Vert\cdot\Vert$ is the $\ell_\infty$ norm then $\imath$ does indeed have dense range.

• On the other hand, if $E=\ell_1$ and $\Vert\cdot\Vert$ is the $\ell_\infty$ norm, then by considering "blocks" which have $\ell_\infty$-norm 1 and large $\ell_1$-norm, we can construct an isometry on $c_0$ which is not approximable by operators of the form $\imath(T)$; in particular, $\imath$ does not have dense range in this case. I can't find the piece of paper where I wrote down the details, but I seem to recall that one obtains the same answer if we replace $\ell_1$ by $\ell_p$ for $1\leq p < \infty$ and take $\Vert\cdot\Vert$ to be the $\ell_r$ norm for any $p<r\leq\infty$.

So here are two explicit questions. I haven't looked at them properly since about 2004/5, so they may well have straightforward solutions.

Q1. Let $\Vert\cdot\Vert$ be any norm on $c_{00}$, and let $F$ be the completion of $c_{00}$ in this norm. Does $\imath: B(c_{00},\Vert\cdot\Vert)\to B(F)$ have dense range?

Q2. Let $(F,\Vert\cdot\Vert)$ be a Banach space with an unconditional basis. Let $E$ be a proper dense subspace of $F$ which is a Banach space under some norm that dominates $\Vert\cdot\Vert$. Does $\imath: B(E,\Vert\cdot\Vert)\to B(F)$ always have non-dense range?

If the answers to these are known, does anyone know where I might find references to these in the literature?

Update 5th July 2010: Q1 has a positive answer, as given by Bill Johnson below (a simmilar approach was also elaborated by Pietro Majer). As pointed out (ibid.) the question can be rephrased/generalized to the following:

given a separable Banach space $F$ and a dense linear subspace $E$ of countable dimension, can every bounded operator on $F$ be approximated by operators which take $E$ to $E$?

I'd still be interested to know the answer to Q2, even in the special cases where $F=\ell_p$ for some $1\leq p < \infty$.

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# Approximating operators on Banach spaces by bounded operators on a proper dense subspace

While digging through old piles of notes and jottings, I came across a question I'd looked at several years ago. While I was able to get partial answers, it seemed even then that the answer should be known and in the literature somewhere, but I never knew where to start looking. So I thought I'd ask here on MO if anyone knows of a reference for this observation.

Here's the notation and background for the question. Let $(E,\Vert\cdot\Vert)$ be a real, normed vector space (I think the complex case works out to be almost identical). We will see shortly that my question is vacuous unless $E$ is incomplete. Let $F$ be the completion of $E$.

Denote by $B(E,\Vert\cdot\Vert)$ the space of all linear maps $T:E\to E$ which ae bounded with respect to $\Vert\cdot\Vert$, i.e. there exists $C$ depending on $T$ such that $$\Vert T(x)\Vert \leq C\Vert x\Vert \;\;\hbox{for all x\in E.}$$

Clearly each $T\in B(E,\vert\cdot\Vert)$ extends uniquely to a bounded linear operator $F\to F$, and we thus get an injective algebra homomorphism $\imath:B(E,\Vert\cdot\Vert)\to B(F)$. The question arises: when does $\imath$ have dense range?

• It is not hard to show that if $E=c_{00}$ and $\Vert\cdot\Vert$ is the $\ell_\infty$ norm then $\imath$ does indeed have dense range.

• On the other hand, if $E=\ell_1$ and $\Vert\cdot\Vert$ is the $\ell_\infty$ norm, then by considering "blocks" which have $\ell_\infty$-norm 1 and large $\ell_1$-norm, we can construct an isometry on $c_0$ which is not approximable by operators of the form $\imath(T)$; in particular, $\imath$ does not have dense range in this case. I can't find the piece of paper where I wrote down the details, but I seem to recall that one obtains the same answer if we replace $\ell_1$ by $\ell_p$ for $1\leq p < \infty$ and take $\Vert\cdot\Vert$ to be the $\ell_r$ norm for any $p<r\leq\infty$.

So here are two explicit questions. I haven't looked at them properly since about 2004/5, so they may well have straightforward solutions.

Q1. Let $\Vert\cdot\Vert$ be any norm on $c_{00}$, and let $F$ be the completion of $c_{00}$ in this norm. Does $\imath: B(c_{00},\Vert\cdot\Vert)\to B(F)$ have dense range?

Q2. Let $(F,\Vert\cdot\Vert)$ be a Banach space with an unconditional basis. Let $E$ be a proper dense subspace of $F$ which is a Banach space under some norm that dominates $\Vert\cdot\Vert$. Does $\imath: B(E,\Vert\cdot\Vert)\to B(F)$ always have non-dense range?

If the answers to these are known, does anyone know where I might find references to these in the literature?