Closed range theorem

In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.

The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Statement

Let $X$ and $Y$ be Banach spaces, $T:D(T)\to Y$ a closed linear operator whose domain $D(T)$ is dense in $X,$ and $T'$ the transpose of $T$ . The theorem asserts that the following conditions are equivalent:

$R(T),$ the range of $T,$ is closed in $Y.$
$R(T'),$ the range of $T',$ is closed in $X',$ the dual of $X.$
$R(T)=N(T')^{\perp }=\left\{y\in Y:\langle x^{*},y\rangle =0\quad {\text{for all}}\quad x^{*}\in N(T')\right\}.$
$R(T')=N(T)^{\perp }=\left\{x^{*}\in X':\langle x^{*},y\rangle =0\quad {\text{for all}}\quad y\in N(T)\right\}.$

Where $N(T)$ and $N(T')$ are the null space of $T$ and $T'$ , respectively.

Note that there is always an inclusion $R(T)\subseteq N(T')^{\perp }$ , because if $y=Tx$ and $x^{*}\in N(T')$ , then $\langle x^{*},y\rangle =\langle T'x^{*},x\rangle =0$ . Likewise, there is an inclusion $R(T')\subseteq N(T)^{\perp }$ . So the non-trivial part of the above theorem is the opposite inclusion in the final two bullets.

Corollaries

Several corollaries are immediate from the theorem. For instance, a densely defined closed operator $T$ as above has $R(T)=Y$ if and only if the transpose $T'$ has a continuous inverse. Similarly, $R(T')=X'$ if and only if $T$ has a continuous inverse.

Sketch of proof

Since the graph of T is closed, the proof reduces to the case when $T:X\to Y$ is a bounded operator between Banach spaces. Now, $T$ factors as $X{\overset {p}{\to }}X/\operatorname {ker} T{\overset {T_{0}}{\to }}\operatorname {im} T{\overset {i}{\hookrightarrow }}Y$ . Dually, $T'$ is

Y'\to (\operatorname {im} T)'{\overset {T_{0}'}{\to }}(X/\operatorname {ker} T)'\to X'.

Now, if $\operatorname {im} T$ is closed, then it is Banach and so by the open mapping theorem, $T_{0}$ is a topological isomorphism. It follows that $T_{0}'$ is an isomorphism and then $\operatorname {im} (T')=\operatorname {ker} (T)^{\bot }$ . (More work is needed for the other implications.) $\square$

References

Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag.

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