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an_ordinary_mathematician
an_ordinary_mathematician
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Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$.

One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?

EDIT: Although the answers cover my original question, i think it is quite natural at this point to ask wether this is true if $H$ and $K$ are more generally Banach space instead of Hilbert spaces.

Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$.

One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?

Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$.

One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?

EDIT: Although the answers cover my original question, i think it is quite natural at this point to ask wether this is true if $H$ and $K$ are more generally Banach space instead of Hilbert spaces.

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an_ordinary_mathematician
an_ordinary_mathematician
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Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$. 

One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?

Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator. One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?

Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$. 

One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?

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an_ordinary_mathematician
an_ordinary_mathematician
  • 2.4k
  • 6
  • 22

Existence of a bounded right inverse to a linear closed surjective operator

Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator. One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id_K$. In fact this is certainly true if $A$ is bijective but the proof doesn't seem to go through with the hypothesis of surjectivity.

Any ideas what is going on in this situation ?