Matrices of bounded linear operators
Let $X,Y$ be Banach spaces and let $A=(A_{n,k})$ be an infinite matrix of bounded linear operators $A_{n,k}:X to Y$. Suppose $sup_n sum_k |A_{n,k}|<infty$.
Property: For each sequence $x=(x_1,x_2,ldots)$ contained in compact of $X$, the image
$$
Ax:=left(sum_k A_{n,k}x_k:nge 1right)
$$
is a well-defined sequence contained in a compact of $Y$. Does such kind of matrices have a name? Or the property is always verified?
(Their property is reminescent of compact operators, i.e., the images of bounded sets are relatively compact.)
banach-spaces compactness compact-operators
asked Jan 4 at 2:37
NduccioNduccio
513313
add a comment |
Let $X,Y$ be Banach spaces and let $A=(A_{n,k})$ be an infinite matrix of bounded linear operators $A_{n,k}:X to Y$. Suppose $sup_n sum_k |A_{n,k}|<infty$.
Property: For each sequence $x=(x_1,x_2,ldots)$ contained in compact of $X$, the image
$$
Ax:=left(sum_k A_{n,k}x_k:nge 1right)
$$
is a well-defined sequence contained in a compact of $Y$. Does such kind of matrices have a name? Or the property is always verified?
(Their property is reminescent of compact operators, i.e., the images of bounded sets are relatively compact.)
banach-spaces compactness compact-operators
asked Jan 4 at 2:37
NduccioNduccio
513313
1
As a side note, such type of matrices appear when checking if a linear map $A:mathcal Atomathcal B$ between $C^*$-algebras $mathcal A,mathcal B$ is completely positive although I feel like this is not what you were looking for.
– Frederik vom Ende
Jan 4 at 9:55
This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences.
– Nduccio
Jan 4 at 10:31
add a comment |
Let $X,Y$ be Banach spaces and let $A=(A_{n,k})$ be an infinite matrix of bounded linear operators $A_{n,k}:X to Y$. Suppose $sup_n sum_k |A_{n,k}|<infty$.
Property: For each sequence $x=(x_1,x_2,ldots)$ contained in compact of $X$, the image
$$
Ax:=left(sum_k A_{n,k}x_k:nge 1right)
$$
is a well-defined sequence contained in a compact of $Y$. Does such kind of matrices have a name? Or the property is always verified?
(Their property is reminescent of compact operators, i.e., the images of bounded sets are relatively compact.)
banach-spaces compactness compact-operators
asked Jan 4 at 2:37
NduccioNduccio
513313
Let $X,Y$ be Banach spaces and let $A=(A_{n,k})$ be an infinite matrix of bounded linear operators $A_{n,k}:X to Y$. Suppose $sup_n sum_k |A_{n,k}|<infty$.
Property: For each sequence $x=(x_1,x_2,ldots)$ contained in compact of $X$, the image
$$
Ax:=left(sum_k A_{n,k}x_k:nge 1right)
$$
is a well-defined sequence contained in a compact of $Y$. Does such kind of matrices have a name? Or the property is always verified?
(Their property is reminescent of compact operators, i.e., the images of bounded sets are relatively compact.)
banach-spaces compactness compact-operators
banach-spaces compactness compact-operators
asked Jan 4 at 2:37
NduccioNduccio
513313
asked Jan 4 at 2:37
NduccioNduccio
513313
asked Jan 4 at 2:37
NduccioNduccio
513313
asked Jan 4 at 2:37
NduccioNduccio
513313
asked Jan 4 at 2:37
NduccioNduccio
513313
513313
1
As a side note, such type of matrices appear when checking if a linear map $A:mathcal Atomathcal B$ between $C^*$-algebras $mathcal A,mathcal B$ is completely positive although I feel like this is not what you were looking for.
– Frederik vom Ende
Jan 4 at 9:55
This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences.
– Nduccio
Jan 4 at 10:31
add a comment |
1
As a side note, such type of matrices appear when checking if a linear map $A:mathcal Atomathcal B$ between $C^*$-algebras $mathcal A,mathcal B$ is completely positive although I feel like this is not what you were looking for.
– Frederik vom Ende
Jan 4 at 9:55
This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences.
– Nduccio
Jan 4 at 10:31
1
1
As a side note, such type of matrices appear when checking if a linear map $A:mathcal Atomathcal B$ between $C^*$-algebras $mathcal A,mathcal B$ is completely positive although I feel like this is not what you were looking for.
– Frederik vom Ende
Jan 4 at 9:55
As a side note, such type of matrices appear when checking if a linear map $A:mathcal Atomathcal B$ between $C^*$-algebras $mathcal A,mathcal B$ is completely positive although I feel like this is not what you were looking for.
– Frederik vom Ende
Jan 4 at 9:55
This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences.
– Nduccio
Jan 4 at 10:31
This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences.
– Nduccio
Jan 4 at 10:31
add a comment |
1 Answer
1
active
oldest
votes
The condition $M:=sup_n sum_k |A_{n,k}|<infty$ is sufficient for $Ax$ well-defined for all $x in ell_infty(X)$. Then the linear map $xmapsto Ax$ on $ell_infty(X)$ is continuous because
$$
|Ax-Ay|=sup_n |sum_k A_{n,k}(x_k-y_k)| le M|x-y|.
$$
To conclude, the continuous image of relatively compact sets is relatively compact, see here.
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
add a comment |
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1 Answer
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1 Answer
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active
oldest
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votes
The condition $M:=sup_n sum_k |A_{n,k}|<infty$ is sufficient for $Ax$ well-defined for all $x in ell_infty(X)$. Then the linear map $xmapsto Ax$ on $ell_infty(X)$ is continuous because
$$
|Ax-Ay|=sup_n |sum_k A_{n,k}(x_k-y_k)| le M|x-y|.
$$
To conclude, the continuous image of relatively compact sets is relatively compact, see here.
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
add a comment |
The condition $M:=sup_n sum_k |A_{n,k}|<infty$ is sufficient for $Ax$ well-defined for all $x in ell_infty(X)$. Then the linear map $xmapsto Ax$ on $ell_infty(X)$ is continuous because
$$
|Ax-Ay|=sup_n |sum_k A_{n,k}(x_k-y_k)| le M|x-y|.
$$
To conclude, the continuous image of relatively compact sets is relatively compact, see here.
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
add a comment |
The condition $M:=sup_n sum_k |A_{n,k}|<infty$ is sufficient for $Ax$ well-defined for all $x in ell_infty(X)$. Then the linear map $xmapsto Ax$ on $ell_infty(X)$ is continuous because
$$
|Ax-Ay|=sup_n |sum_k A_{n,k}(x_k-y_k)| le M|x-y|.
$$
To conclude, the continuous image of relatively compact sets is relatively compact, see here.
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
The condition $M:=sup_n sum_k |A_{n,k}|<infty$ is sufficient for $Ax$ well-defined for all $x in ell_infty(X)$. Then the linear map $xmapsto Ax$ on $ell_infty(X)$ is continuous because
$$
|Ax-Ay|=sup_n |sum_k A_{n,k}(x_k-y_k)| le M|x-y|.
$$
To conclude, the continuous image of relatively compact sets is relatively compact, see here.
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
answered Jan 4 at 2:51
Paolo LeonettiPaolo Leonetti
11.4k21550
11.4k21550
add a comment |
add a comment |
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As a side note, such type of matrices appear when checking if a linear map $A:mathcal Atomathcal B$ between $C^*$-algebras $mathcal A,mathcal B$ is completely positive although I feel like this is not what you were looking for.
– Frederik vom Ende
Jan 4 at 9:55
This is not a homework task, yes I am interested. How is it related to "complete regularity" of such matrices? By the name I guess this is stronger than the classical Toeplitz conditions of mapping convergent sequences into convergent sequences.
– Nduccio
Jan 4 at 10:31