Mixing
Adjoint of a polynomial in a closed linear operator.
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Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T:
$ P(T) := prod_{i=1}^{n}(T - lambda_i I)^{m_i} $ and we recall that $ D(P(T)) = D(T^{m_1 + m_2.....+ m_n})$. It is known that $ P(T) $ is closed and densely defined. Also, one can easily see that $ P(T') subset P(T)' $, where $ T'$ is the adjoint of $ T$ . Is it true that $ P(T') = P(T)' $?
functional-analysis operator-theory adjoint-operators
edited Jan 19 at 10:45
MaoWao
3,568617
asked Jan 19 at 10:28
GAyoubGAyoub
162
$endgroup$
add a comment |
$begingroup$
Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T:
$ P(T) := prod_{i=1}^{n}(T - lambda_i I)^{m_i} $ and we recall that $ D(P(T)) = D(T^{m_1 + m_2.....+ m_n})$. It is known that $ P(T) $ is closed and densely defined. Also, one can easily see that $ P(T') subset P(T)' $, where $ T'$ is the adjoint of $ T$ . Is it true that $ P(T') = P(T)' $?
functional-analysis operator-theory adjoint-operators
edited Jan 19 at 10:45
MaoWao
3,568617
asked Jan 19 at 10:28
GAyoubGAyoub
162
$endgroup$
$begingroup$
Hi, welcome to MSE. I added some tags to your question in order for it to get some more exposure. In general it is advisable to not only tag a specific topic (like "adjoint-operators"), but also broader field.
$endgroup$
– MaoWao
Jan 19 at 10:48
$begingroup$
Thank you MaoWao.
$endgroup$
– GAyoub
Jan 19 at 11:13
add a comment |
$begingroup$
Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T:
$ P(T) := prod_{i=1}^{n}(T - lambda_i I)^{m_i} $ and we recall that $ D(P(T)) = D(T^{m_1 + m_2.....+ m_n})$. It is known that $ P(T) $ is closed and densely defined. Also, one can easily see that $ P(T') subset P(T)' $, where $ T'$ is the adjoint of $ T$ . Is it true that $ P(T') = P(T)' $?
functional-analysis operator-theory adjoint-operators
edited Jan 19 at 10:45
MaoWao
3,568617
asked Jan 19 at 10:28
GAyoubGAyoub
162
$endgroup$
Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T:
$ P(T) := prod_{i=1}^{n}(T - lambda_i I)^{m_i} $ and we recall that $ D(P(T)) = D(T^{m_1 + m_2.....+ m_n})$. It is known that $ P(T) $ is closed and densely defined. Also, one can easily see that $ P(T') subset P(T)' $, where $ T'$ is the adjoint of $ T$ . Is it true that $ P(T') = P(T)' $?
functional-analysis operator-theory adjoint-operators
functional-analysis operator-theory adjoint-operators
edited Jan 19 at 10:45
MaoWao
3,568617
asked Jan 19 at 10:28
GAyoubGAyoub
162
edited Jan 19 at 10:45
MaoWao
3,568617
asked Jan 19 at 10:28
GAyoubGAyoub
162
edited Jan 19 at 10:45
MaoWao
3,568617
edited Jan 19 at 10:45
MaoWao
3,568617
edited Jan 19 at 10:45
MaoWao
3,568617
3,568617
asked Jan 19 at 10:28
GAyoubGAyoub
162
asked Jan 19 at 10:28
GAyoubGAyoub
162
asked Jan 19 at 10:28
GAyoubGAyoub
162
162
$begingroup$
Hi, welcome to MSE. I added some tags to your question in order for it to get some more exposure. In general it is advisable to not only tag a specific topic (like "adjoint-operators"), but also broader field.
$endgroup$
– MaoWao
Jan 19 at 10:48
$begingroup$
Thank you MaoWao.
$endgroup$
– GAyoub
Jan 19 at 11:13
add a comment |
$begingroup$
Hi, welcome to MSE. I added some tags to your question in order for it to get some more exposure. In general it is advisable to not only tag a specific topic (like "adjoint-operators"), but also broader field.
$endgroup$
– MaoWao
Jan 19 at 10:48
$begingroup$
Thank you MaoWao.
$endgroup$
– GAyoub
Jan 19 at 11:13
$begingroup$
Hi, welcome to MSE. I added some tags to your question in order for it to get some more exposure. In general it is advisable to not only tag a specific topic (like "adjoint-operators"), but also broader field.
$endgroup$
– MaoWao
Jan 19 at 10:48
$begingroup$
Hi, welcome to MSE. I added some tags to your question in order for it to get some more exposure. In general it is advisable to not only tag a specific topic (like "adjoint-operators"), but also broader field.
$endgroup$
– MaoWao
Jan 19 at 10:48
$begingroup$
Thank you MaoWao.
$endgroup$
– GAyoub
Jan 19 at 11:13
$begingroup$
Thank you MaoWao.
$endgroup$
– GAyoub
Jan 19 at 11:13
add a comment |
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$begingroup$
Hi, welcome to MSE. I added some tags to your question in order for it to get some more exposure. In general it is advisable to not only tag a specific topic (like "adjoint-operators"), but also broader field.
$endgroup$
– MaoWao
Jan 19 at 10:48
$begingroup$
Thank you MaoWao.
$endgroup$
– GAyoub
Jan 19 at 11:13