Does convergence in Hilbert-Schmidt norm imply convergence of singular values?
$begingroup$
Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = sum_n lambda_n u_n otimes v_n$.
Now let $A_i$ be a sequence of operators converging to $A$ in Hilbert-Schmidt norm. Do the singular values of the $A_i$ converge to the singular values of $A$, and if so in what precise sense? What can we say about the singular vectors?
Finally, does this require that $H_1$ and $H_2$ be separable?
functional-analysis operator-theory hilbert-spaces spectral-theory singularvalues
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
$endgroup$
add a comment |
$begingroup$
Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = sum_n lambda_n u_n otimes v_n$.
Now let $A_i$ be a sequence of operators converging to $A$ in Hilbert-Schmidt norm. Do the singular values of the $A_i$ converge to the singular values of $A$, and if so in what precise sense? What can we say about the singular vectors?
Finally, does this require that $H_1$ and $H_2$ be separable?
functional-analysis operator-theory hilbert-spaces spectral-theory singularvalues
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
$endgroup$
add a comment |
$begingroup$
Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = sum_n lambda_n u_n otimes v_n$.
Now let $A_i$ be a sequence of operators converging to $A$ in Hilbert-Schmidt norm. Do the singular values of the $A_i$ converge to the singular values of $A$, and if so in what precise sense? What can we say about the singular vectors?
Finally, does this require that $H_1$ and $H_2$ be separable?
functional-analysis operator-theory hilbert-spaces spectral-theory singularvalues
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
$endgroup$
Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = sum_n lambda_n u_n otimes v_n$.
Now let $A_i$ be a sequence of operators converging to $A$ in Hilbert-Schmidt norm. Do the singular values of the $A_i$ converge to the singular values of $A$, and if so in what precise sense? What can we say about the singular vectors?
Finally, does this require that $H_1$ and $H_2$ be separable?
functional-analysis operator-theory hilbert-spaces spectral-theory singularvalues
functional-analysis operator-theory hilbert-spaces spectral-theory singularvalues
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
asked Dec 16 '18 at 9:58
NeuromathNeuromath
14710
14710
add a comment |
add a comment |
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