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BAYMAX
BAYMAX
10:34 AM
.
I was thinking how the map $T \rightarrow T^{*}$ is linear?
where $T :X \rightarrow Y$ isa alinear map
and $T^{*} : Y' \rightarrow X'$ is the adjoint operator map
 
Martin Sleziak
Martin Sleziak
I guess that notation $T\mapsto T^*$is a bit less confusing.
 
BAYMAX
BAYMAX
in fact we define $(T^{*}(f))(x) = f(T(x))$
 
Martin Sleziak
Martin Sleziak
All you need to do is to verify that $(S+T)^*=S^*+T^*$ and $(kT)^*=kT^*$.
I'd say that both follow almost immediately if you rewrite the definition.
 
BAYMAX
BAYMAX
$((S+T)^{*}(f))(x) = f((S+T)(x))$
we know that $S,T$ are continuous
 
Martin Sleziak
Martin Sleziak
I don't think you need continuity here. You could get analogous thing for algebraic dual.
 
BAYMAX
BAYMAX
10:38 AM
so $(S+T)(x) = S(x) + T(x)$
oh k
so $f(S(x) + T(x)) = f(S(x)) +f(T(x)) = (T^*(f))(x) + (S^*(f))(x)$
and
for showing $(kT)^{} = kT^{}$
why is latex not displaying to the power *
 
Martin Sleziak
Martin Sleziak
@BAYMAX I'll just add that this leads you to $(S+T)^*(f)=S^*(f)+T^*(f)$ for each $f$ and eventually to $(S+T)^*=S^*+T^*$.
 
BAYMAX
BAYMAX
myn was also right?
i just forgot to say $\forall f$
 
Martin Sleziak
Martin Sleziak
@BAYMAX I'd say it was right, just unfinished.
 
BAYMAX
BAYMAX
and $\forall x \in X$
 
Martin Sleziak
Martin Sleziak
Depending how detailed you want to make the argument.
 
BAYMAX
BAYMAX
10:43 AM
ok
the posts
 
Martin Sleziak
Martin Sleziak
@BAYMAX My best guess is that sometimes the star is understood as sign for italics.
 
BAYMAX
BAYMAX
showing $||T|| = ||T^*||$ the MSE posts are not so much answered!
 
Martin Sleziak
Martin Sleziak
The MSE posts? Which ones?
 
BAYMAX
BAYMAX
like this one
 
Martin Sleziak
Martin Sleziak
@BAYMAX That one is closed.
 
BAYMAX
BAYMAX
10:46 AM
i can view it
?
 
Martin Sleziak
Martin Sleziak
Yes, so you can probably see that it is closed.
There is a big duplicate banner at the top.
You're interested only in Hilbert case?
 
BAYMAX
BAYMAX
hmm..in my proof i see no usage of Hilbert space
in my proof means in my note
 
Martin Sleziak
Martin Sleziak
In any case, there are probably many posts about this on the main site. You could try searching in Approach0 for this or for this.
@BAYMAX The post you linked is asking explicitly about Hilbert spaces.
I have tried to search for posts tagged adjoint+norm but there are only a few of them.
 
BAYMAX
BAYMAX
ok
so we cannot prove them withtout using HBET
 
Martin Sleziak
Martin Sleziak
Anyway, this is more question about searching on the site, so it would probably be more suitable for another chatroom.
 
 
2 hours later…
BAYMAX
BAYMAX
12:31 PM
well lets prove $||T^*|| \leq ||T||$
$||(T^*(f))(x)|| \leq ||T^*|| ||f|| ||x||$
so $sup_{||x|| = 1} ||(T^*(f))(x)|| \leq ||T^*|| ||f||$
or
$||T^*(f)|| \leq ||T|| ||f||$
$sup(\frac{||T^*(f)||}{||f||})$ for $||f|| = 1$
we get $||T^*||$
so $||T^*|| \leq ||T||$
next we prove $||T^*|| \geq ||T||$
How we get this -
$(T^* f)(x_{0})(x_{0}) = f(T(x_{0})) = ||T(x_{0})||$
Yes,as a corollary to HBET there exists $f \in Y'$ such that $f(T(x_{0})) = ||T(x_{0})||$
and $||f|| = 1$
$||T|| = sup\{T(x) | ||x|| \leq 1 \}$
so then for $\epsilon > 0 $thereexists $x_{0} \in X$ such that $||T|| - \epsilon < ||T(x)||$
so $||T|| - \epsilon <|| T^* f||$
so $||T|| - \epsilon < ||T^*||$
as $\epsilon $ is arbitray
so $||T|| \leq ||T^*||$
and hence $||T|| = ||T^*||$
 
 
4 hours later…
BAYMAX
BAYMAX
5:19 PM
.
Let $T \in BL(H)$ where $H$ is a Hilbert space
for fixed $y \in H$ , define $f_{y} : H \rightarrow K$ by $f_{y} = <Tx,y>$
is $f_{y}$ a bounded linear map on $H $ ?
$||f_{y}(x)|| = ||<Tx,y>|| \leq ||Tx|| . ||y|| \leq ||T|| ||x|| ||y||$
Now $||f_{y}|| = sup\{f_{y}(x) : ||x|| \leq 1\}$
so $||f_{y}|| \leq ||T|| . ||y||$
 
 
1 hour later…
Feeds
Feeds
6:59 PM
2
Q: Hilbert valued martingales - help with reference

LucioI'm currently studying the theory of SPDEs on the book "Stochastic equations in infinite dimensions" by da Prato, Zabczyk. In the book, the theory of stochastic processes with values on a Banach space $E$, and in the particular the notion of $E$-valued brownian motion, is also introduced. When th...

functional-analysis banach-spaces stochastic-integrals stochastic-analysis stochastic-pde
 

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