Mixing
The Hilbert space $textIm(M^1/2)$, where $M$ is a positive operator.
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In this question $F$ stands for a complex Hilbert space with inner product $langlecdot;,;cdotrangle$ and the norm $|cdot|$. Let $mathcalB(F)$ the algebra of all bounded linear operators on $F$.
Let $Min mathcalB(F)^+$ (i.e. $langle Mx;, ;xranglegeq 0$ for all $xin F$) and $P$ be the orthogonal projection of $F$ onto the closure of $textIm(M)$.
According to this answer $textIm(M^1/2)$ endow with the inner product
$$tag1 (M^1/2x,M^1/2y)_textIm(M^1/2):=langle Px, Pyrangle,;forall, x,y in F,$$
is a Hilbert space.
َAccording also to this answer, for any $xin F$ there exists a sequence $(x_n)_n$ with
$$M^1/2x=lim_nto infty Mx_n Longleftrightarrow lim_nto infty|Mx_n-M^1/2x|=0.$$
If $SinmathcalB(textIm(M^1/2))$, is it true that for all $xin F$, we have
$$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);?$$
I ask this question because I see in this paper that $textIm(M)$ is dense in $textIm(M^1/2)$. However, I'm facing difficulties to understand if the density is with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1) or with respect to the topology induced by $langlecdot,cdotrangle$.
functional-analysis operator-theory hilbert-spaces
asked Jul 28 at 11:29
Schüler
1,3311321
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up vote
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In this question $F$ stands for a complex Hilbert space with inner product $langlecdot;,;cdotrangle$ and the norm $|cdot|$. Let $mathcalB(F)$ the algebra of all bounded linear operators on $F$.
Let $Min mathcalB(F)^+$ (i.e. $langle Mx;, ;xranglegeq 0$ for all $xin F$) and $P$ be the orthogonal projection of $F$ onto the closure of $textIm(M)$.
According to this answer $textIm(M^1/2)$ endow with the inner product
$$tag1 (M^1/2x,M^1/2y)_textIm(M^1/2):=langle Px, Pyrangle,;forall, x,y in F,$$
is a Hilbert space.
َAccording also to this answer, for any $xin F$ there exists a sequence $(x_n)_n$ with
$$M^1/2x=lim_nto infty Mx_n Longleftrightarrow lim_nto infty|Mx_n-M^1/2x|=0.$$
If $SinmathcalB(textIm(M^1/2))$, is it true that for all $xin F$, we have
$$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);?$$
I ask this question because I see in this paper that $textIm(M)$ is dense in $textIm(M^1/2)$. However, I'm facing difficulties to understand if the density is with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1) or with respect to the topology induced by $langlecdot,cdotrangle$.
functional-analysis operator-theory hilbert-spaces
asked Jul 28 at 11:29
Schüler
1,3311321
The second link shows that $operatornameIm(M)$ is dense in $operatornameIm(M^1/2)$ with respect to the subspace topology induced by $F$, i.e. with respect to $langle,cdot,,,cdot,rangle$. I don't know right away whether it's also dense with respect to the topology coming from $(1)$. For a general $S in mathcalB(F)$, we must expect $SM^1/2xnotin operatornameIm(M^1/2)$, so the highlighted part doesn't make sense in the stated generality.
– Daniel Fischer♦
17 hours ago
@DanielFischer Thank you very much for your comment. Sorry, here $SinmathcalB(textIm(M^1/2))$ and thus $SM^1/2xin textIm(M^1/2)$ for all $xin F$.
– Schüler
15 hours ago
@DanielFischer Is the equality $$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);$$ true only if $textIm(M)$ is dense in $textIm(M^1/2)$ with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1)? Thank you for your help.
– Schüler
12 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
In this question $F$ stands for a complex Hilbert space with inner product $langlecdot;,;cdotrangle$ and the norm $|cdot|$. Let $mathcalB(F)$ the algebra of all bounded linear operators on $F$.
Let $Min mathcalB(F)^+$ (i.e. $langle Mx;, ;xranglegeq 0$ for all $xin F$) and $P$ be the orthogonal projection of $F$ onto the closure of $textIm(M)$.
According to this answer $textIm(M^1/2)$ endow with the inner product
$$tag1 (M^1/2x,M^1/2y)_textIm(M^1/2):=langle Px, Pyrangle,;forall, x,y in F,$$
is a Hilbert space.
َAccording also to this answer, for any $xin F$ there exists a sequence $(x_n)_n$ with
$$M^1/2x=lim_nto infty Mx_n Longleftrightarrow lim_nto infty|Mx_n-M^1/2x|=0.$$
If $SinmathcalB(textIm(M^1/2))$, is it true that for all $xin F$, we have
$$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);?$$
I ask this question because I see in this paper that $textIm(M)$ is dense in $textIm(M^1/2)$. However, I'm facing difficulties to understand if the density is with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1) or with respect to the topology induced by $langlecdot,cdotrangle$.
functional-analysis operator-theory hilbert-spaces
asked Jul 28 at 11:29
Schüler
1,3311321
In this question $F$ stands for a complex Hilbert space with inner product $langlecdot;,;cdotrangle$ and the norm $|cdot|$. Let $mathcalB(F)$ the algebra of all bounded linear operators on $F$.
Let $Min mathcalB(F)^+$ (i.e. $langle Mx;, ;xranglegeq 0$ for all $xin F$) and $P$ be the orthogonal projection of $F$ onto the closure of $textIm(M)$.
According to this answer $textIm(M^1/2)$ endow with the inner product
$$tag1 (M^1/2x,M^1/2y)_textIm(M^1/2):=langle Px, Pyrangle,;forall, x,y in F,$$
is a Hilbert space.
َAccording also to this answer, for any $xin F$ there exists a sequence $(x_n)_n$ with
$$M^1/2x=lim_nto infty Mx_n Longleftrightarrow lim_nto infty|Mx_n-M^1/2x|=0.$$
If $SinmathcalB(textIm(M^1/2))$, is it true that for all $xin F$, we have
$$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);?$$
I ask this question because I see in this paper that $textIm(M)$ is dense in $textIm(M^1/2)$. However, I'm facing difficulties to understand if the density is with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1) or with respect to the topology induced by $langlecdot,cdotrangle$.
functional-analysis operator-theory hilbert-spaces
asked Jul 28 at 11:29
Schüler
1,3311321
edited 15 hours ago
asked Jul 28 at 11:29
Schüler
1,3311321
asked Jul 28 at 11:29
Schüler
1,3311321
asked Jul 28 at 11:29
Schüler
1,3311321
1,3311321
The second link shows that $operatornameIm(M)$ is dense in $operatornameIm(M^1/2)$ with respect to the subspace topology induced by $F$, i.e. with respect to $langle,cdot,,,cdot,rangle$. I don't know right away whether it's also dense with respect to the topology coming from $(1)$. For a general $S in mathcalB(F)$, we must expect $SM^1/2xnotin operatornameIm(M^1/2)$, so the highlighted part doesn't make sense in the stated generality.
– Daniel Fischer♦
17 hours ago
@DanielFischer Thank you very much for your comment. Sorry, here $SinmathcalB(textIm(M^1/2))$ and thus $SM^1/2xin textIm(M^1/2)$ for all $xin F$.
– Schüler
15 hours ago
@DanielFischer Is the equality $$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);$$ true only if $textIm(M)$ is dense in $textIm(M^1/2)$ with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1)? Thank you for your help.
– Schüler
12 hours ago
add a comment |Â
The second link shows that $operatornameIm(M)$ is dense in $operatornameIm(M^1/2)$ with respect to the subspace topology induced by $F$, i.e. with respect to $langle,cdot,,,cdot,rangle$. I don't know right away whether it's also dense with respect to the topology coming from $(1)$. For a general $S in mathcalB(F)$, we must expect $SM^1/2xnotin operatornameIm(M^1/2)$, so the highlighted part doesn't make sense in the stated generality.
– Daniel Fischer♦
17 hours ago
@DanielFischer Thank you very much for your comment. Sorry, here $SinmathcalB(textIm(M^1/2))$ and thus $SM^1/2xin textIm(M^1/2)$ for all $xin F$.
– Schüler
15 hours ago
@DanielFischer Is the equality $$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);$$ true only if $textIm(M)$ is dense in $textIm(M^1/2)$ with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1)? Thank you for your help.
– Schüler
12 hours ago
The second link shows that $operatornameIm(M)$ is dense in $operatornameIm(M^1/2)$ with respect to the subspace topology induced by $F$, i.e. with respect to $langle,cdot,,,cdot,rangle$. I don't know right away whether it's also dense with respect to the topology coming from $(1)$. For a general $S in mathcalB(F)$, we must expect $SM^1/2xnotin operatornameIm(M^1/2)$, so the highlighted part doesn't make sense in the stated generality.
– Daniel Fischer♦
17 hours ago
The second link shows that $operatornameIm(M)$ is dense in $operatornameIm(M^1/2)$ with respect to the subspace topology induced by $F$, i.e. with respect to $langle,cdot,,,cdot,rangle$. I don't know right away whether it's also dense with respect to the topology coming from $(1)$. For a general $S in mathcalB(F)$, we must expect $SM^1/2xnotin operatornameIm(M^1/2)$, so the highlighted part doesn't make sense in the stated generality.
– Daniel Fischer♦
17 hours ago
@DanielFischer Thank you very much for your comment. Sorry, here $SinmathcalB(textIm(M^1/2))$ and thus $SM^1/2xin textIm(M^1/2)$ for all $xin F$.
– Schüler
15 hours ago
@DanielFischer Thank you very much for your comment. Sorry, here $SinmathcalB(textIm(M^1/2))$ and thus $SM^1/2xin textIm(M^1/2)$ for all $xin F$.
– Schüler
15 hours ago
@DanielFischer Is the equality $$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);$$ true only if $textIm(M)$ is dense in $textIm(M^1/2)$ with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1)? Thank you for your help.
– Schüler
12 hours ago
@DanielFischer Is the equality $$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);$$ true only if $textIm(M)$ is dense in $textIm(M^1/2)$ with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1)? Thank you for your help.
– Schüler
12 hours ago
add a comment |Â
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The second link shows that $operatornameIm(M)$ is dense in $operatornameIm(M^1/2)$ with respect to the subspace topology induced by $F$, i.e. with respect to $langle,cdot,,,cdot,rangle$. I don't know right away whether it's also dense with respect to the topology coming from $(1)$. For a general $S in mathcalB(F)$, we must expect $SM^1/2xnotin operatornameIm(M^1/2)$, so the highlighted part doesn't make sense in the stated generality.
– Daniel Fischer♦
17 hours ago
@DanielFischer Thank you very much for your comment. Sorry, here $SinmathcalB(textIm(M^1/2))$ and thus $SM^1/2xin textIm(M^1/2)$ for all $xin F$.
– Schüler
15 hours ago
@DanielFischer Is the equality $$|SM^1/2x|_textIm(M^1/2)=lim_nto infty |SMx_n|_textIm(M^1/2);$$ true only if $textIm(M)$ is dense in $textIm(M^1/2)$ with respect to the topology induced by the inner product on $textIm(M^1/2)$ defined in (1)? Thank you for your help.
– Schüler
12 hours ago