Can be the kernel infinite dimensional?
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Let $T:Hto H$ be a continuous, compact, self adjoint linear operator on the Hilbert space H. Can we have $dim ker(T)=infty$?
functional-analysis
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Let $T:Hto H$ be a continuous, compact, self adjoint linear operator on the Hilbert space H. Can we have $dim ker(T)=infty$?
functional-analysis
Did you intend to ask: "Can we have $dim (ker(T))<infty$?
â Kavi Rama Murthy
Aug 29 at 8:10
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up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $T:Hto H$ be a continuous, compact, self adjoint linear operator on the Hilbert space H. Can we have $dim ker(T)=infty$?
functional-analysis
Let $T:Hto H$ be a continuous, compact, self adjoint linear operator on the Hilbert space H. Can we have $dim ker(T)=infty$?
functional-analysis
edited Aug 29 at 6:37
Arthur
101k795176
101k795176
asked Aug 29 at 6:34
Bogdan
63649
63649
Did you intend to ask: "Can we have $dim (ker(T))<infty$?
â Kavi Rama Murthy
Aug 29 at 8:10
add a comment |Â
Did you intend to ask: "Can we have $dim (ker(T))<infty$?
â Kavi Rama Murthy
Aug 29 at 8:10
Did you intend to ask: "Can we have $dim (ker(T))<infty$?
â Kavi Rama Murthy
Aug 29 at 8:10
Did you intend to ask: "Can we have $dim (ker(T))<infty$?
â Kavi Rama Murthy
Aug 29 at 8:10
add a comment |Â
1 Answer
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oldest
votes
up vote
4
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The zero map (given by $T(x)=0$) is a compact, continuous, self-adjoint linear operator with $dimker$ equal to the dimension of $H$. So, if $H$ is infinite-dimensional...
For a non-trivial example, take, for instance, orthogonal projection onto a line: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
@uniquesolution sure
â Arthur
Aug 29 at 7:14
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
The zero map (given by $T(x)=0$) is a compact, continuous, self-adjoint linear operator with $dimker$ equal to the dimension of $H$. So, if $H$ is infinite-dimensional...
For a non-trivial example, take, for instance, orthogonal projection onto a line: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
@uniquesolution sure
â Arthur
Aug 29 at 7:14
add a comment |Â
up vote
4
down vote
accepted
The zero map (given by $T(x)=0$) is a compact, continuous, self-adjoint linear operator with $dimker$ equal to the dimension of $H$. So, if $H$ is infinite-dimensional...
For a non-trivial example, take, for instance, orthogonal projection onto a line: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
@uniquesolution sure
â Arthur
Aug 29 at 7:14
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
The zero map (given by $T(x)=0$) is a compact, continuous, self-adjoint linear operator with $dimker$ equal to the dimension of $H$. So, if $H$ is infinite-dimensional...
For a non-trivial example, take, for instance, orthogonal projection onto a line: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.
The zero map (given by $T(x)=0$) is a compact, continuous, self-adjoint linear operator with $dimker$ equal to the dimension of $H$. So, if $H$ is infinite-dimensional...
For a non-trivial example, take, for instance, orthogonal projection onto a line: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.
edited Aug 29 at 7:15
answered Aug 29 at 6:40
Arthur
101k795176
101k795176
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
@uniquesolution sure
â Arthur
Aug 29 at 7:14
add a comment |Â
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
@uniquesolution sure
â Arthur
Aug 29 at 7:14
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
Can you also give a non-trivial example?
â uniquesolution
Aug 29 at 7:11
@uniquesolution sure
â Arthur
Aug 29 at 7:14
@uniquesolution sure
â Arthur
Aug 29 at 7:14
add a comment |Â
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Did you intend to ask: "Can we have $dim (ker(T))<infty$?
â Kavi Rama Murthy
Aug 29 at 8:10