When does the spectrum of an element in a Banach algebra with involution lie in the open right half-plane?
$begingroup$
Let $A$ be a Banach algebra with involution, $xin A$ and $tin
{mathbb R}$ such that $t>rho(xx^*)$. Show that $sigma(te-xx^*)$
lies in the open right half-plane.
I have no idea! It's obvious that $t-rho(xx^*)>0$. Maybe we should use this to
conclude $rho(te-xx^*)>0$ and then getting the statement! Would you
please help me with that?
spectral-theory banach-algebras
asked Dec 25 '18 at 21:34
SsFfSsFf
485
$endgroup$
add a comment |
$begingroup$
Let $A$ be a Banach algebra with involution, $xin A$ and $tin
{mathbb R}$ such that $t>rho(xx^*)$. Show that $sigma(te-xx^*)$
lies in the open right half-plane.
I have no idea! It's obvious that $t-rho(xx^*)>0$. Maybe we should use this to
conclude $rho(te-xx^*)>0$ and then getting the statement! Would you
please help me with that?
spectral-theory banach-algebras
asked Dec 25 '18 at 21:34
SsFfSsFf
485
$endgroup$
1
$begingroup$
Is $e$ the identity element of $A$?
$endgroup$
– Math1000
Dec 25 '18 at 22:02
add a comment |
$begingroup$
Let $A$ be a Banach algebra with involution, $xin A$ and $tin
{mathbb R}$ such that $t>rho(xx^*)$. Show that $sigma(te-xx^*)$
lies in the open right half-plane.
I have no idea! It's obvious that $t-rho(xx^*)>0$. Maybe we should use this to
conclude $rho(te-xx^*)>0$ and then getting the statement! Would you
please help me with that?
spectral-theory banach-algebras
asked Dec 25 '18 at 21:34
SsFfSsFf
485
$endgroup$
Let $A$ be a Banach algebra with involution, $xin A$ and $tin
{mathbb R}$ such that $t>rho(xx^*)$. Show that $sigma(te-xx^*)$
lies in the open right half-plane.
I have no idea! It's obvious that $t-rho(xx^*)>0$. Maybe we should use this to
conclude $rho(te-xx^*)>0$ and then getting the statement! Would you
please help me with that?
spectral-theory banach-algebras
spectral-theory banach-algebras
asked Dec 25 '18 at 21:34
SsFfSsFf
485
asked Dec 25 '18 at 21:34
SsFfSsFf
485
asked Dec 25 '18 at 21:34
SsFfSsFf
485
asked Dec 25 '18 at 21:34
SsFfSsFf
485
asked Dec 25 '18 at 21:34
SsFfSsFf
485
485
1
$begingroup$
Is $e$ the identity element of $A$?
$endgroup$
– Math1000
Dec 25 '18 at 22:02
add a comment |
1
$begingroup$
Is $e$ the identity element of $A$?
$endgroup$
– Math1000
Dec 25 '18 at 22:02
1
1
$begingroup$
Is $e$ the identity element of $A$?
$endgroup$
– Math1000
Dec 25 '18 at 22:02
$begingroup$
Is $e$ the identity element of $A$?
$endgroup$
– Math1000
Dec 25 '18 at 22:02
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I think you have the right idea. We can replace $xx^*$ with $y$, as the anatomy of $xx^*$ is irrelevant.
Note that,
begin{align*}
&s in sigma(y) \
iff , &(y - se)^{-1} text{ doesn't exist} \
iff , &(se - y)^{-1} text{ doesn't exist} \
iff , &(te - y - (t - s)e)^{-1} text{ doesn't exist} \
iff , &t - s in sigma(te - y).
end{align*}
If $s in sigma(y)$, then
$$Re(t - s) = t - Re(s) ge t - |s| > 0.$$
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
$endgroup$
add a comment |
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$begingroup$
I think you have the right idea. We can replace $xx^*$ with $y$, as the anatomy of $xx^*$ is irrelevant.
Note that,
begin{align*}
&s in sigma(y) \
iff , &(y - se)^{-1} text{ doesn't exist} \
iff , &(se - y)^{-1} text{ doesn't exist} \
iff , &(te - y - (t - s)e)^{-1} text{ doesn't exist} \
iff , &t - s in sigma(te - y).
end{align*}
If $s in sigma(y)$, then
$$Re(t - s) = t - Re(s) ge t - |s| > 0.$$
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
$endgroup$
add a comment |
$begingroup$
I think you have the right idea. We can replace $xx^*$ with $y$, as the anatomy of $xx^*$ is irrelevant.
Note that,
begin{align*}
&s in sigma(y) \
iff , &(y - se)^{-1} text{ doesn't exist} \
iff , &(se - y)^{-1} text{ doesn't exist} \
iff , &(te - y - (t - s)e)^{-1} text{ doesn't exist} \
iff , &t - s in sigma(te - y).
end{align*}
If $s in sigma(y)$, then
$$Re(t - s) = t - Re(s) ge t - |s| > 0.$$
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
$endgroup$
add a comment |
$begingroup$
I think you have the right idea. We can replace $xx^*$ with $y$, as the anatomy of $xx^*$ is irrelevant.
Note that,
begin{align*}
&s in sigma(y) \
iff , &(y - se)^{-1} text{ doesn't exist} \
iff , &(se - y)^{-1} text{ doesn't exist} \
iff , &(te - y - (t - s)e)^{-1} text{ doesn't exist} \
iff , &t - s in sigma(te - y).
end{align*}
If $s in sigma(y)$, then
$$Re(t - s) = t - Re(s) ge t - |s| > 0.$$
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
$endgroup$
I think you have the right idea. We can replace $xx^*$ with $y$, as the anatomy of $xx^*$ is irrelevant.
Note that,
begin{align*}
&s in sigma(y) \
iff , &(y - se)^{-1} text{ doesn't exist} \
iff , &(se - y)^{-1} text{ doesn't exist} \
iff , &(te - y - (t - s)e)^{-1} text{ doesn't exist} \
iff , &t - s in sigma(te - y).
end{align*}
If $s in sigma(y)$, then
$$Re(t - s) = t - Re(s) ge t - |s| > 0.$$
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
answered Dec 25 '18 at 23:23
Theo BenditTheo Bendit
19.4k12353
19.4k12353
add a comment |
add a comment |
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1
$begingroup$
Is $e$ the identity element of $A$?
$endgroup$
– Math1000
Dec 25 '18 at 22:02