Connection between the eigenfunctions of the compact operators $T[f](xin H_1)=int_{H_1}k(x,y)f(y)dy$ and...
$begingroup$
Let $H_1$ and $H_2$ be Hilbert spaces.
Suppose we have a compact integral operator $T:H_1 to H_1$ given by
$$
T[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_1.
$$
Suppose we also have a compact integral operator $R:H_1 to H_2$ given by
$$
R[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_2.
$$
As $R$ and $T$ are compact we have spectral decompositions
$$
T = sum_{n=0}^infty lambda_n^T langle phi_n^T,cdot rangle phi_n^T, quad quad R = sum_{n=0}^infty lambda_n^R langle phi_n^R,cdot rangle psi_n^R.
$$
As $T$ maps from $H_1$ to itself it can be written in terms of the eigenfunctions $phi_n^T in H_1$, whereas $R$ maps from $H_1$ to to $H_2$ so it is written in terms of the eigenfunctions $phi_n^R in H_1$ and $psi_n^R in H_2$.
As $R$ and $T$ are are very similar operators, I would like to know whether there is a connection between the eigenfunctions $phi_n^T$ and $phi_n^R$ which are both defined on $H_1$. For instance, can one of these functions be written in terms of the other?
functional-analysis operator-theory hilbert-spaces spectral-theory integral-equations
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
$endgroup$
add a comment |
$begingroup$
Let $H_1$ and $H_2$ be Hilbert spaces.
Suppose we have a compact integral operator $T:H_1 to H_1$ given by
$$
T[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_1.
$$
Suppose we also have a compact integral operator $R:H_1 to H_2$ given by
$$
R[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_2.
$$
As $R$ and $T$ are compact we have spectral decompositions
$$
T = sum_{n=0}^infty lambda_n^T langle phi_n^T,cdot rangle phi_n^T, quad quad R = sum_{n=0}^infty lambda_n^R langle phi_n^R,cdot rangle psi_n^R.
$$
As $T$ maps from $H_1$ to itself it can be written in terms of the eigenfunctions $phi_n^T in H_1$, whereas $R$ maps from $H_1$ to to $H_2$ so it is written in terms of the eigenfunctions $phi_n^R in H_1$ and $psi_n^R in H_2$.
As $R$ and $T$ are are very similar operators, I would like to know whether there is a connection between the eigenfunctions $phi_n^T$ and $phi_n^R$ which are both defined on $H_1$. For instance, can one of these functions be written in terms of the other?
functional-analysis operator-theory hilbert-spaces spectral-theory integral-equations
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
$endgroup$
$begingroup$
There are several issues with your question, that make it hard to understand what you are trying to ask. First, while it is possible to integrate over a Hilbert space in some sense, I doubt that's what you are trying to do; you write as if $f$ is a function with domain $H_1$, which I don't think is what you want. Second, you write as if ${phi_n^T}$ and ${psi_n^R}$ are orthonormal bases (are they?), in which case it seems you are assuming that your operators are selfadjoint; but you didn't say so. Third, did you really want $phi_n^R$?
$endgroup$
– Martin Argerami
Dec 13 '18 at 15:03
add a comment |
$begingroup$
Let $H_1$ and $H_2$ be Hilbert spaces.
Suppose we have a compact integral operator $T:H_1 to H_1$ given by
$$
T[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_1.
$$
Suppose we also have a compact integral operator $R:H_1 to H_2$ given by
$$
R[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_2.
$$
As $R$ and $T$ are compact we have spectral decompositions
$$
T = sum_{n=0}^infty lambda_n^T langle phi_n^T,cdot rangle phi_n^T, quad quad R = sum_{n=0}^infty lambda_n^R langle phi_n^R,cdot rangle psi_n^R.
$$
As $T$ maps from $H_1$ to itself it can be written in terms of the eigenfunctions $phi_n^T in H_1$, whereas $R$ maps from $H_1$ to to $H_2$ so it is written in terms of the eigenfunctions $phi_n^R in H_1$ and $psi_n^R in H_2$.
As $R$ and $T$ are are very similar operators, I would like to know whether there is a connection between the eigenfunctions $phi_n^T$ and $phi_n^R$ which are both defined on $H_1$. For instance, can one of these functions be written in terms of the other?
functional-analysis operator-theory hilbert-spaces spectral-theory integral-equations
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
$endgroup$
Let $H_1$ and $H_2$ be Hilbert spaces.
Suppose we have a compact integral operator $T:H_1 to H_1$ given by
$$
T[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_1.
$$
Suppose we also have a compact integral operator $R:H_1 to H_2$ given by
$$
R[f](x) = int_{H_1} k(x,y)f(y)dy, quad quad x in H_2.
$$
As $R$ and $T$ are compact we have spectral decompositions
$$
T = sum_{n=0}^infty lambda_n^T langle phi_n^T,cdot rangle phi_n^T, quad quad R = sum_{n=0}^infty lambda_n^R langle phi_n^R,cdot rangle psi_n^R.
$$
As $T$ maps from $H_1$ to itself it can be written in terms of the eigenfunctions $phi_n^T in H_1$, whereas $R$ maps from $H_1$ to to $H_2$ so it is written in terms of the eigenfunctions $phi_n^R in H_1$ and $psi_n^R in H_2$.
As $R$ and $T$ are are very similar operators, I would like to know whether there is a connection between the eigenfunctions $phi_n^T$ and $phi_n^R$ which are both defined on $H_1$. For instance, can one of these functions be written in terms of the other?
functional-analysis operator-theory hilbert-spaces spectral-theory integral-equations
functional-analysis operator-theory hilbert-spaces spectral-theory integral-equations
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
asked Dec 13 '18 at 8:28
sonicboomsonicboom
3,73082853
3,73082853
$begingroup$
There are several issues with your question, that make it hard to understand what you are trying to ask. First, while it is possible to integrate over a Hilbert space in some sense, I doubt that's what you are trying to do; you write as if $f$ is a function with domain $H_1$, which I don't think is what you want. Second, you write as if ${phi_n^T}$ and ${psi_n^R}$ are orthonormal bases (are they?), in which case it seems you are assuming that your operators are selfadjoint; but you didn't say so. Third, did you really want $phi_n^R$?
$endgroup$
– Martin Argerami
Dec 13 '18 at 15:03
add a comment |
$begingroup$
There are several issues with your question, that make it hard to understand what you are trying to ask. First, while it is possible to integrate over a Hilbert space in some sense, I doubt that's what you are trying to do; you write as if $f$ is a function with domain $H_1$, which I don't think is what you want. Second, you write as if ${phi_n^T}$ and ${psi_n^R}$ are orthonormal bases (are they?), in which case it seems you are assuming that your operators are selfadjoint; but you didn't say so. Third, did you really want $phi_n^R$?
$endgroup$
– Martin Argerami
Dec 13 '18 at 15:03
$begingroup$
There are several issues with your question, that make it hard to understand what you are trying to ask. First, while it is possible to integrate over a Hilbert space in some sense, I doubt that's what you are trying to do; you write as if $f$ is a function with domain $H_1$, which I don't think is what you want. Second, you write as if ${phi_n^T}$ and ${psi_n^R}$ are orthonormal bases (are they?), in which case it seems you are assuming that your operators are selfadjoint; but you didn't say so. Third, did you really want $phi_n^R$?
$endgroup$
– Martin Argerami
Dec 13 '18 at 15:03
$begingroup$
There are several issues with your question, that make it hard to understand what you are trying to ask. First, while it is possible to integrate over a Hilbert space in some sense, I doubt that's what you are trying to do; you write as if $f$ is a function with domain $H_1$, which I don't think is what you want. Second, you write as if ${phi_n^T}$ and ${psi_n^R}$ are orthonormal bases (are they?), in which case it seems you are assuming that your operators are selfadjoint; but you didn't say so. Third, did you really want $phi_n^R$?
$endgroup$
– Martin Argerami
Dec 13 '18 at 15:03
add a comment |
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$begingroup$
There are several issues with your question, that make it hard to understand what you are trying to ask. First, while it is possible to integrate over a Hilbert space in some sense, I doubt that's what you are trying to do; you write as if $f$ is a function with domain $H_1$, which I don't think is what you want. Second, you write as if ${phi_n^T}$ and ${psi_n^R}$ are orthonormal bases (are they?), in which case it seems you are assuming that your operators are selfadjoint; but you didn't say so. Third, did you really want $phi_n^R$?
$endgroup$
– Martin Argerami
Dec 13 '18 at 15:03