Question on extending densely defined linear operators on Hilbert space

Question

Let $H$ be a Hilbert space.

Is there a theorem that states that for a densely defined continuous linear operator $T: D(T) \subset H \to H$ there exists a unique continuous linear extension to $H$?

And furthermore,

If $T$ is an isometry then so is its extension?

I tried to find something about this and I found here on Wikipedia that this is true for an abstract Wiener space. But I don't know about Wiener spaces and I think there should be a more basic proof because Hilbert spaces are already nice spaces.

If the answer is affirmative to either of my question could please either provide a reference to a book or similar where I can study the proof or else post a proof in an answer?

score 1 · Answer 1 · answered Feb 15 '15 at 01:39

1

You can use the continuous linear extension theorem, which is slightly more general than you need it to be.

answered Feb 15 '15 at 01:39

Ben Grossmann

225,327

Thanks and what about the isometric case? And the article you link to does not appear to contain a proof. I'd appreciate a proof or reference to a proof. – Feb 15 '15 at 02:00
For example: Introductory Functional Analysis with Applications - Kreyszig; p. 99 (Theorem 2.7-11). – Ben Grossmann Feb 15 '15 at 02:11
See a link to the book here. – Ben Grossmann Feb 15 '15 at 02:12
Thank you, I have the book. Also, the isometric case follows from this case. – Feb 15 '15 at 08:16

Question on extending densely defined linear operators on Hilbert space

1 Answers1