Normal operator with real spectrum is hermitian.
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Consider a normal operator on a complex Hilbert space with its spectrum contained in the real line. Show that the operator is hermitian without using the spectral theorem.
What I have tried- Since the spectrum is inside the real line, the boundary of the spectrum as a subset of the complex numbers is the spectrum itself. Hence the spectrum is equal to the approximate point spectrum. I have been unable to proceed further.
functional-analysis operator-theory self-adjoint-operators
asked Dec 2 at 13:12
Manan
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up vote
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Consider a normal operator on a complex Hilbert space with its spectrum contained in the real line. Show that the operator is hermitian without using the spectral theorem.
What I have tried- Since the spectrum is inside the real line, the boundary of the spectrum as a subset of the complex numbers is the spectrum itself. Hence the spectrum is equal to the approximate point spectrum. I have been unable to proceed further.
functional-analysis operator-theory self-adjoint-operators
asked Dec 2 at 13:12
Manan
1,075611
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider a normal operator on a complex Hilbert space with its spectrum contained in the real line. Show that the operator is hermitian without using the spectral theorem.
What I have tried- Since the spectrum is inside the real line, the boundary of the spectrum as a subset of the complex numbers is the spectrum itself. Hence the spectrum is equal to the approximate point spectrum. I have been unable to proceed further.
functional-analysis operator-theory self-adjoint-operators
asked Dec 2 at 13:12
Manan
1,075611
Consider a normal operator on a complex Hilbert space with its spectrum contained in the real line. Show that the operator is hermitian without using the spectral theorem.
What I have tried- Since the spectrum is inside the real line, the boundary of the spectrum as a subset of the complex numbers is the spectrum itself. Hence the spectrum is equal to the approximate point spectrum. I have been unable to proceed further.
functional-analysis operator-theory self-adjoint-operators
functional-analysis operator-theory self-adjoint-operators
asked Dec 2 at 13:12
Manan
1,075611
asked Dec 2 at 13:12
Manan
1,075611
edited Dec 2 at 13:19
asked Dec 2 at 13:12
Manan
1,075611
asked Dec 2 at 13:12
Manan
1,075611
asked Dec 2 at 13:12
Manan
1,075611
1,075611
add a comment |
add a comment |
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