Mixing
Orthogonal projection in Hilbert space
$begingroup$
Projector $Pneq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions:
a) $P$ is self-adjoint, $P=P^*$
b) $P$ is normal, i.e. $P^*P=PP^*$
c) $P$ is positive
d) $|P|=1$
By definition, if $Pneq 0$ is orthogonal projector that is equivalent with a).
$a)implies b):$ $P^*P=P^2=PP^*$
$a)iff c):$ $P$ is positive if it's square form is nonnegative. That is equivalent with $langle Pf,frangle$ is real. Operator is self adjoint iff it's square form is real.
Can anyone give me a hint how to prove rest of implications.
functional-analysis orthonormal projection
edited Jan 12 at 22:17
Bernard
121k740116
asked Jan 12 at 22:11
HanaHana
161
$endgroup$
add a comment |
$begingroup$
Projector $Pneq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions:
a) $P$ is self-adjoint, $P=P^*$
b) $P$ is normal, i.e. $P^*P=PP^*$
c) $P$ is positive
d) $|P|=1$
By definition, if $Pneq 0$ is orthogonal projector that is equivalent with a).
$a)implies b):$ $P^*P=P^2=PP^*$
$a)iff c):$ $P$ is positive if it's square form is nonnegative. That is equivalent with $langle Pf,frangle$ is real. Operator is self adjoint iff it's square form is real.
Can anyone give me a hint how to prove rest of implications.
functional-analysis orthonormal projection
edited Jan 12 at 22:17
Bernard
121k740116
asked Jan 12 at 22:11
HanaHana
161
$endgroup$
$begingroup$
Topology and Modern Analysis by George Simmons is a good reference for this. The proofs are somewhat long and the result is available in many books. Your proof of a) iff c) is not correct.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 23:29
$begingroup$
In the mentioned book I couldn't find the proofs. Can you tell me other references?
$endgroup$
– Hana
Jan 14 at 11:46
add a comment |
$begingroup$
Projector $Pneq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions:
a) $P$ is self-adjoint, $P=P^*$
b) $P$ is normal, i.e. $P^*P=PP^*$
c) $P$ is positive
d) $|P|=1$
By definition, if $Pneq 0$ is orthogonal projector that is equivalent with a).
$a)implies b):$ $P^*P=P^2=PP^*$
$a)iff c):$ $P$ is positive if it's square form is nonnegative. That is equivalent with $langle Pf,frangle$ is real. Operator is self adjoint iff it's square form is real.
Can anyone give me a hint how to prove rest of implications.
functional-analysis orthonormal projection
edited Jan 12 at 22:17
Bernard
121k740116
asked Jan 12 at 22:11
HanaHana
161
$endgroup$
Projector $Pneq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions:
a) $P$ is self-adjoint, $P=P^*$
b) $P$ is normal, i.e. $P^*P=PP^*$
c) $P$ is positive
d) $|P|=1$
By definition, if $Pneq 0$ is orthogonal projector that is equivalent with a).
$a)implies b):$ $P^*P=P^2=PP^*$
$a)iff c):$ $P$ is positive if it's square form is nonnegative. That is equivalent with $langle Pf,frangle$ is real. Operator is self adjoint iff it's square form is real.
Can anyone give me a hint how to prove rest of implications.
functional-analysis orthonormal projection
functional-analysis orthonormal projection
edited Jan 12 at 22:17
Bernard
121k740116
asked Jan 12 at 22:11
HanaHana
161
edited Jan 12 at 22:17
Bernard
121k740116
asked Jan 12 at 22:11
HanaHana
161
edited Jan 12 at 22:17
Bernard
121k740116
edited Jan 12 at 22:17
Bernard
121k740116
edited Jan 12 at 22:17
Bernard
121k740116
121k740116
asked Jan 12 at 22:11
HanaHana
161
asked Jan 12 at 22:11
HanaHana
161
asked Jan 12 at 22:11
HanaHana
161
161
$begingroup$
Topology and Modern Analysis by George Simmons is a good reference for this. The proofs are somewhat long and the result is available in many books. Your proof of a) iff c) is not correct.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 23:29
$begingroup$
In the mentioned book I couldn't find the proofs. Can you tell me other references?
$endgroup$
– Hana
Jan 14 at 11:46
add a comment |
$begingroup$
Topology and Modern Analysis by George Simmons is a good reference for this. The proofs are somewhat long and the result is available in many books. Your proof of a) iff c) is not correct.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 23:29
$begingroup$
In the mentioned book I couldn't find the proofs. Can you tell me other references?
$endgroup$
– Hana
Jan 14 at 11:46
$begingroup$
Topology and Modern Analysis by George Simmons is a good reference for this. The proofs are somewhat long and the result is available in many books. Your proof of a) iff c) is not correct.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 23:29
$begingroup$
Topology and Modern Analysis by George Simmons is a good reference for this. The proofs are somewhat long and the result is available in many books. Your proof of a) iff c) is not correct.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 23:29
$begingroup$
In the mentioned book I couldn't find the proofs. Can you tell me other references?
$endgroup$
– Hana
Jan 14 at 11:46
$begingroup$
In the mentioned book I couldn't find the proofs. Can you tell me other references?
$endgroup$
– Hana
Jan 14 at 11:46
add a comment |
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$begingroup$
Topology and Modern Analysis by George Simmons is a good reference for this. The proofs are somewhat long and the result is available in many books. Your proof of a) iff c) is not correct.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 23:29
$begingroup$
In the mentioned book I couldn't find the proofs. Can you tell me other references?
$endgroup$
– Hana
Jan 14 at 11:46