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$T^\times$ and $S^\times$ are the adjoint operators of $T,S\in B(X,Y)$, $X$ and $Y$ normed spaces. $T^\times$ and $S^\times$ are defined on the dual spaces which contain the ranges of $T$ and $S$, respectively.

Prove $(S+T)^\times = S^\times +T^\times$.

I'm having some troubles understanding this section. The homework contains a bunch of 'left-to-the-reader' exercises. This is one that wasn't assigned. If someone could help me through this one, that'd be great.

Desperate Fluffy
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1 Answers1

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What does $(S+T)^\times$ do? Well, if $f\in Y^*$ and $x\in X$ then

$$\langle (S+T)^\times f, x \rangle = \langle f, (S+T)x\rangle.$$

However, $$\langle f, (S+T)x\rangle = \langle f, Sx + Tx\rangle = \langle f, Sx \rangle + \langle f, Tx\rangle = \langle S^\times f, x\rangle + \langle T^\times f, x\rangle.$$

Tomasz Kania
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  • This is not necessarily a Hilbert space. If you mean the dual pairing you should say so for clarity, if not then this is an incomplete answer. – Adam Hughes Nov 13 '14 at 18:16
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    Adam, this is the standard symbol for duality between a Banach space and its dual. Btw. I am a Banach-space person. – Tomasz Kania Nov 13 '14 at 18:16
  • It's not uncommon, but it's not standardized either. The applied maths course at my university, for example, does not have this anywhere in the text and never makes reference to it. It's common, but it's not universal. I mention it because it may be used by Banach space people, but not all courses for students use it (that's my point, sorry if that was unclear). In other words the op might not recognize it (or he might). – Adam Hughes Nov 13 '14 at 18:17
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    There are very few notations in mathematics which merit the designation "universal". In my experience, the $\langle \cdot, \cdot \rangle$ notation in the context of normed space duality is common enough not to require further explanation. Regards. – Robert Lewis Nov 13 '14 at 18:40
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    @RobertLewis I agree from a mathematical standpoint, but I'm speaking to pedagogy. If the student has not seen it (which is entirely conceivable in this case) it could prove confusing. Professionals have seen it, I agree, but it's not always taught this way to students in classes. (I'm assuming "in my experience" means in your experience with the subject, not with the teaching of it) – Adam Hughes Nov 13 '14 at 19:00
  • What notation do you then suggest? The OP must have certainly seen the definition of $T^\times$, hence he/she is familiar with at least one way of writing this down. – Tomasz Kania Nov 13 '14 at 19:06
  • Notation was a little unfamiliar (my prof has been using just $()$ not $\langle\rangle$), but I can puzzle through notation. Thanks! ^_^ – Desperate Fluffy Nov 13 '14 at 21:09