Mixing
Showing convexity of a set in $mathbb{C}^k$
$begingroup$
Let $mathcal{H}$ be an infinite dimensional separable Hilbert Space and $Tinmathscr{B(mathcal{H})}$. Suppose $$mu_k(T):=left{
begin{pmatrix}
lambda_1 \
lambda_2 \
vdots \
lambda_k
end{pmatrix}inmathbb{C}^k: PTP=
begin{pmatrix}
lambda_1 & 0 & cdots & 0 \
0 & lambda_2 &cdots & 0 \
vdots & vdots & ddots\
0 & 0 & cdots & lambda_k \
end{pmatrix}, text{ for some orthogonal projection $P$ of rank } k right}$$
Is $mu_k(T)$ convex in $mathbb{C}^k$?
Comments: I can only see that $mu_k(T)$ is non-empty but I could not construct desired projection of rank k to show convexity of $mu_k(T).$
Any hint/comment regarding convexity of $mu_k(T)$ is highly appreciated. Thanks in advance.
functional-analysis analysis operator-theory operator-algebras matrix-analysis
asked Jan 22 at 9:34
PikuPiku
685
$endgroup$
|
show 3 more comments
$begingroup$
Let $mathcal{H}$ be an infinite dimensional separable Hilbert Space and $Tinmathscr{B(mathcal{H})}$. Suppose $$mu_k(T):=left{
begin{pmatrix}
lambda_1 \
lambda_2 \
vdots \
lambda_k
end{pmatrix}inmathbb{C}^k: PTP=
begin{pmatrix}
lambda_1 & 0 & cdots & 0 \
0 & lambda_2 &cdots & 0 \
vdots & vdots & ddots\
0 & 0 & cdots & lambda_k \
end{pmatrix}, text{ for some orthogonal projection $P$ of rank } k right}$$
Is $mu_k(T)$ convex in $mathbb{C}^k$?
Comments: I can only see that $mu_k(T)$ is non-empty but I could not construct desired projection of rank k to show convexity of $mu_k(T).$
Any hint/comment regarding convexity of $mu_k(T)$ is highly appreciated. Thanks in advance.
functional-analysis analysis operator-theory operator-algebras matrix-analysis
asked Jan 22 at 9:34
PikuPiku
685
$endgroup$
$begingroup$
Well, I don't know, but the spectrum needs not be convex. These sets look like a kind of spectrum so probably they are not convex as well.
$endgroup$
– Giuseppe Negro
Jan 22 at 9:38
1
$begingroup$
It's not clear how your set is defined. What's a matrix in $B(H)$? Do you want $PTP=sum_jlambda Q_j$ for fixed pairwise orthogonal rank-one projections, or for any such family of projections?
$endgroup$
– Martin Argerami
Jan 23 at 1:52
1
$begingroup$
@GiuseppeNegro: it looks like a numerical range, and those are usually convex.
$endgroup$
– Martin Argerami
Jan 23 at 19:40
1
$begingroup$
Something seems off in your definition; if $P$ is a rank-$k$ projection (acting on $mathcal H$) then $PTPinmathcal B(mathcal H)$ but the right-hand side is a matrix. I assume you want $P$ to be some sort of compression (?), i.e. for the standard basis $(hat e_i)_{i=1}^k$ in $mathbb C^k$ and an arbitrary orthonormal system $(e_i)_{i=1}^k$ in $mathcal H$ define $$P:mathbb C^ktomathcal Hquad hat e_imapsto e_i$$ for all $i=1,ldots,k$ (plus linear extension onto all of $mathbb C^k$). This way $P^dagger TPinmathbb C^{ktimes k}$ with fixed basis.
$endgroup$
– Frederik vom Ende
Jan 24 at 15:01
1
$begingroup$
(2/2) In that case, the object in question is close to (but slightly more general than) the rank-$k$ numerical range. Maybe this connection alone is of use to you already?
$endgroup$
– Frederik vom Ende
Jan 24 at 15:06
|
show 3 more comments
$begingroup$
Let $mathcal{H}$ be an infinite dimensional separable Hilbert Space and $Tinmathscr{B(mathcal{H})}$. Suppose $$mu_k(T):=left{
begin{pmatrix}
lambda_1 \
lambda_2 \
vdots \
lambda_k
end{pmatrix}inmathbb{C}^k: PTP=
begin{pmatrix}
lambda_1 & 0 & cdots & 0 \
0 & lambda_2 &cdots & 0 \
vdots & vdots & ddots\
0 & 0 & cdots & lambda_k \
end{pmatrix}, text{ for some orthogonal projection $P$ of rank } k right}$$
Is $mu_k(T)$ convex in $mathbb{C}^k$?
Comments: I can only see that $mu_k(T)$ is non-empty but I could not construct desired projection of rank k to show convexity of $mu_k(T).$
Any hint/comment regarding convexity of $mu_k(T)$ is highly appreciated. Thanks in advance.
functional-analysis analysis operator-theory operator-algebras matrix-analysis
asked Jan 22 at 9:34
PikuPiku
685
$endgroup$
Let $mathcal{H}$ be an infinite dimensional separable Hilbert Space and $Tinmathscr{B(mathcal{H})}$. Suppose $$mu_k(T):=left{
begin{pmatrix}
lambda_1 \
lambda_2 \
vdots \
lambda_k
end{pmatrix}inmathbb{C}^k: PTP=
begin{pmatrix}
lambda_1 & 0 & cdots & 0 \
0 & lambda_2 &cdots & 0 \
vdots & vdots & ddots\
0 & 0 & cdots & lambda_k \
end{pmatrix}, text{ for some orthogonal projection $P$ of rank } k right}$$
Is $mu_k(T)$ convex in $mathbb{C}^k$?
Comments: I can only see that $mu_k(T)$ is non-empty but I could not construct desired projection of rank k to show convexity of $mu_k(T).$
Any hint/comment regarding convexity of $mu_k(T)$ is highly appreciated. Thanks in advance.
functional-analysis analysis operator-theory operator-algebras matrix-analysis
functional-analysis analysis operator-theory operator-algebras matrix-analysis
asked Jan 22 at 9:34
PikuPiku
685
asked Jan 22 at 9:34
PikuPiku
685
asked Jan 22 at 9:34
PikuPiku
685
asked Jan 22 at 9:34
PikuPiku
685
asked Jan 22 at 9:34
PikuPiku
685
685
$begingroup$
Well, I don't know, but the spectrum needs not be convex. These sets look like a kind of spectrum so probably they are not convex as well.
$endgroup$
– Giuseppe Negro
Jan 22 at 9:38
1
$begingroup$
It's not clear how your set is defined. What's a matrix in $B(H)$? Do you want $PTP=sum_jlambda Q_j$ for fixed pairwise orthogonal rank-one projections, or for any such family of projections?
$endgroup$
– Martin Argerami
Jan 23 at 1:52
1
$begingroup$
@GiuseppeNegro: it looks like a numerical range, and those are usually convex.
$endgroup$
– Martin Argerami
Jan 23 at 19:40
1
$begingroup$
Something seems off in your definition; if $P$ is a rank-$k$ projection (acting on $mathcal H$) then $PTPinmathcal B(mathcal H)$ but the right-hand side is a matrix. I assume you want $P$ to be some sort of compression (?), i.e. for the standard basis $(hat e_i)_{i=1}^k$ in $mathbb C^k$ and an arbitrary orthonormal system $(e_i)_{i=1}^k$ in $mathcal H$ define $$P:mathbb C^ktomathcal Hquad hat e_imapsto e_i$$ for all $i=1,ldots,k$ (plus linear extension onto all of $mathbb C^k$). This way $P^dagger TPinmathbb C^{ktimes k}$ with fixed basis.
$endgroup$
– Frederik vom Ende
Jan 24 at 15:01
1
$begingroup$
(2/2) In that case, the object in question is close to (but slightly more general than) the rank-$k$ numerical range. Maybe this connection alone is of use to you already?
$endgroup$
– Frederik vom Ende
Jan 24 at 15:06
|
show 3 more comments
$begingroup$
Well, I don't know, but the spectrum needs not be convex. These sets look like a kind of spectrum so probably they are not convex as well.
$endgroup$
– Giuseppe Negro
Jan 22 at 9:38
1
$begingroup$
It's not clear how your set is defined. What's a matrix in $B(H)$? Do you want $PTP=sum_jlambda Q_j$ for fixed pairwise orthogonal rank-one projections, or for any such family of projections?
$endgroup$
– Martin Argerami
Jan 23 at 1:52
1
$begingroup$
@GiuseppeNegro: it looks like a numerical range, and those are usually convex.
$endgroup$
– Martin Argerami
Jan 23 at 19:40
1
$begingroup$
Something seems off in your definition; if $P$ is a rank-$k$ projection (acting on $mathcal H$) then $PTPinmathcal B(mathcal H)$ but the right-hand side is a matrix. I assume you want $P$ to be some sort of compression (?), i.e. for the standard basis $(hat e_i)_{i=1}^k$ in $mathbb C^k$ and an arbitrary orthonormal system $(e_i)_{i=1}^k$ in $mathcal H$ define $$P:mathbb C^ktomathcal Hquad hat e_imapsto e_i$$ for all $i=1,ldots,k$ (plus linear extension onto all of $mathbb C^k$). This way $P^dagger TPinmathbb C^{ktimes k}$ with fixed basis.
$endgroup$
– Frederik vom Ende
Jan 24 at 15:01
1
$begingroup$
(2/2) In that case, the object in question is close to (but slightly more general than) the rank-$k$ numerical range. Maybe this connection alone is of use to you already?
$endgroup$
– Frederik vom Ende
Jan 24 at 15:06
$begingroup$
Well, I don't know, but the spectrum needs not be convex. These sets look like a kind of spectrum so probably they are not convex as well.
$endgroup$
– Giuseppe Negro
Jan 22 at 9:38
$begingroup$
Well, I don't know, but the spectrum needs not be convex. These sets look like a kind of spectrum so probably they are not convex as well.
$endgroup$
– Giuseppe Negro
Jan 22 at 9:38
1
1
$begingroup$
It's not clear how your set is defined. What's a matrix in $B(H)$? Do you want $PTP=sum_jlambda Q_j$ for fixed pairwise orthogonal rank-one projections, or for any such family of projections?
$endgroup$
– Martin Argerami
Jan 23 at 1:52
$begingroup$
It's not clear how your set is defined. What's a matrix in $B(H)$? Do you want $PTP=sum_jlambda Q_j$ for fixed pairwise orthogonal rank-one projections, or for any such family of projections?
$endgroup$
– Martin Argerami
Jan 23 at 1:52
1
1
$begingroup$
@GiuseppeNegro: it looks like a numerical range, and those are usually convex.
$endgroup$
– Martin Argerami
Jan 23 at 19:40
$begingroup$
@GiuseppeNegro: it looks like a numerical range, and those are usually convex.
$endgroup$
– Martin Argerami
Jan 23 at 19:40
1
1
$begingroup$
Something seems off in your definition; if $P$ is a rank-$k$ projection (acting on $mathcal H$) then $PTPinmathcal B(mathcal H)$ but the right-hand side is a matrix. I assume you want $P$ to be some sort of compression (?), i.e. for the standard basis $(hat e_i)_{i=1}^k$ in $mathbb C^k$ and an arbitrary orthonormal system $(e_i)_{i=1}^k$ in $mathcal H$ define $$P:mathbb C^ktomathcal Hquad hat e_imapsto e_i$$ for all $i=1,ldots,k$ (plus linear extension onto all of $mathbb C^k$). This way $P^dagger TPinmathbb C^{ktimes k}$ with fixed basis.
$endgroup$
– Frederik vom Ende
Jan 24 at 15:01
$begingroup$
Something seems off in your definition; if $P$ is a rank-$k$ projection (acting on $mathcal H$) then $PTPinmathcal B(mathcal H)$ but the right-hand side is a matrix. I assume you want $P$ to be some sort of compression (?), i.e. for the standard basis $(hat e_i)_{i=1}^k$ in $mathbb C^k$ and an arbitrary orthonormal system $(e_i)_{i=1}^k$ in $mathcal H$ define $$P:mathbb C^ktomathcal Hquad hat e_imapsto e_i$$ for all $i=1,ldots,k$ (plus linear extension onto all of $mathbb C^k$). This way $P^dagger TPinmathbb C^{ktimes k}$ with fixed basis.
$endgroup$
– Frederik vom Ende
Jan 24 at 15:01
1
1
$begingroup$
(2/2) In that case, the object in question is close to (but slightly more general than) the rank-$k$ numerical range. Maybe this connection alone is of use to you already?
$endgroup$
– Frederik vom Ende
Jan 24 at 15:06
$begingroup$
(2/2) In that case, the object in question is close to (but slightly more general than) the rank-$k$ numerical range. Maybe this connection alone is of use to you already?
$endgroup$
– Frederik vom Ende
Jan 24 at 15:06
|
show 3 more comments
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$begingroup$
Well, I don't know, but the spectrum needs not be convex. These sets look like a kind of spectrum so probably they are not convex as well.
$endgroup$
– Giuseppe Negro
Jan 22 at 9:38
1
$begingroup$
It's not clear how your set is defined. What's a matrix in $B(H)$? Do you want $PTP=sum_jlambda Q_j$ for fixed pairwise orthogonal rank-one projections, or for any such family of projections?
$endgroup$
– Martin Argerami
Jan 23 at 1:52
1
$begingroup$
@GiuseppeNegro: it looks like a numerical range, and those are usually convex.
$endgroup$
– Martin Argerami
Jan 23 at 19:40
1
$begingroup$
Something seems off in your definition; if $P$ is a rank-$k$ projection (acting on $mathcal H$) then $PTPinmathcal B(mathcal H)$ but the right-hand side is a matrix. I assume you want $P$ to be some sort of compression (?), i.e. for the standard basis $(hat e_i)_{i=1}^k$ in $mathbb C^k$ and an arbitrary orthonormal system $(e_i)_{i=1}^k$ in $mathcal H$ define $$P:mathbb C^ktomathcal Hquad hat e_imapsto e_i$$ for all $i=1,ldots,k$ (plus linear extension onto all of $mathbb C^k$). This way $P^dagger TPinmathbb C^{ktimes k}$ with fixed basis.
$endgroup$
– Frederik vom Ende
Jan 24 at 15:01
1
$begingroup$
(2/2) In that case, the object in question is close to (but slightly more general than) the rank-$k$ numerical range. Maybe this connection alone is of use to you already?
$endgroup$
– Frederik vom Ende
Jan 24 at 15:06