Set of injective operators is a dense residual set in $mathcal{B}(mathfrak{X})$
$begingroup$
Let $mathfrak{X}$ be a Banach space and $mathcal{B}(mathfrak{X})$ the set of bounded linear operators mapping $mathfrak{X}$ to $mathfrak{X}$. In [1] below it is shown that the set of invertible operators is not necessarily dense in $mathcal{B}(mathfrak{X})$, although it is well known that it is an open set. In [2], it is shown that the set of injections (I suppose we may call this $mathcal{I}(mathfrak{X})$) is not necessarily open in $mathcal{B}(mathfrak{X})$.
My question is on how "common" injectiveness is in $mathcal{B}(mathfrak{X})$. In particular, the results above leave open the possibility that $mathcal{I}(mathfrak{X})$ is a dense $G_delta$ set, and so topologically generic. In either case, it would be interesting to know whether density or the $G_delta$ condition hold. I believe that the spectral theorem already implies that normal operators on Hilbert spaces may be approximated by injective operators (see for instance [3]). Any results that you guys know of would be appreciated!
[1] Showing that $mathcal{G}(ell_2)$ is not dense in $mathcal{B}(ell_2)$ via the right shift
[2] Do bounded linear operators on a Banach space which are injective or have dense range form an open subspace?
[3] The set of invertible normal operator is dense in the set of normal operator
functional-analysis operator-theory banach-spaces
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
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add a comment |
$begingroup$
Let $mathfrak{X}$ be a Banach space and $mathcal{B}(mathfrak{X})$ the set of bounded linear operators mapping $mathfrak{X}$ to $mathfrak{X}$. In [1] below it is shown that the set of invertible operators is not necessarily dense in $mathcal{B}(mathfrak{X})$, although it is well known that it is an open set. In [2], it is shown that the set of injections (I suppose we may call this $mathcal{I}(mathfrak{X})$) is not necessarily open in $mathcal{B}(mathfrak{X})$.
My question is on how "common" injectiveness is in $mathcal{B}(mathfrak{X})$. In particular, the results above leave open the possibility that $mathcal{I}(mathfrak{X})$ is a dense $G_delta$ set, and so topologically generic. In either case, it would be interesting to know whether density or the $G_delta$ condition hold. I believe that the spectral theorem already implies that normal operators on Hilbert spaces may be approximated by injective operators (see for instance [3]). Any results that you guys know of would be appreciated!
[1] Showing that $mathcal{G}(ell_2)$ is not dense in $mathcal{B}(ell_2)$ via the right shift
[2] Do bounded linear operators on a Banach space which are injective or have dense range form an open subspace?
[3] The set of invertible normal operator is dense in the set of normal operator
functional-analysis operator-theory banach-spaces
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
$endgroup$
add a comment |
$begingroup$
Let $mathfrak{X}$ be a Banach space and $mathcal{B}(mathfrak{X})$ the set of bounded linear operators mapping $mathfrak{X}$ to $mathfrak{X}$. In [1] below it is shown that the set of invertible operators is not necessarily dense in $mathcal{B}(mathfrak{X})$, although it is well known that it is an open set. In [2], it is shown that the set of injections (I suppose we may call this $mathcal{I}(mathfrak{X})$) is not necessarily open in $mathcal{B}(mathfrak{X})$.
My question is on how "common" injectiveness is in $mathcal{B}(mathfrak{X})$. In particular, the results above leave open the possibility that $mathcal{I}(mathfrak{X})$ is a dense $G_delta$ set, and so topologically generic. In either case, it would be interesting to know whether density or the $G_delta$ condition hold. I believe that the spectral theorem already implies that normal operators on Hilbert spaces may be approximated by injective operators (see for instance [3]). Any results that you guys know of would be appreciated!
[1] Showing that $mathcal{G}(ell_2)$ is not dense in $mathcal{B}(ell_2)$ via the right shift
[2] Do bounded linear operators on a Banach space which are injective or have dense range form an open subspace?
[3] The set of invertible normal operator is dense in the set of normal operator
functional-analysis operator-theory banach-spaces
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
$endgroup$
Let $mathfrak{X}$ be a Banach space and $mathcal{B}(mathfrak{X})$ the set of bounded linear operators mapping $mathfrak{X}$ to $mathfrak{X}$. In [1] below it is shown that the set of invertible operators is not necessarily dense in $mathcal{B}(mathfrak{X})$, although it is well known that it is an open set. In [2], it is shown that the set of injections (I suppose we may call this $mathcal{I}(mathfrak{X})$) is not necessarily open in $mathcal{B}(mathfrak{X})$.
My question is on how "common" injectiveness is in $mathcal{B}(mathfrak{X})$. In particular, the results above leave open the possibility that $mathcal{I}(mathfrak{X})$ is a dense $G_delta$ set, and so topologically generic. In either case, it would be interesting to know whether density or the $G_delta$ condition hold. I believe that the spectral theorem already implies that normal operators on Hilbert spaces may be approximated by injective operators (see for instance [3]). Any results that you guys know of would be appreciated!
[1] Showing that $mathcal{G}(ell_2)$ is not dense in $mathcal{B}(ell_2)$ via the right shift
[2] Do bounded linear operators on a Banach space which are injective or have dense range form an open subspace?
[3] The set of invertible normal operator is dense in the set of normal operator
functional-analysis operator-theory banach-spaces
functional-analysis operator-theory banach-spaces
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
asked Dec 4 '18 at 20:17
Ikebf Ikebf
233
233
add a comment |
add a comment |
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