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 $dim ker(T)=infty$?







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  • Did you intend to ask: "Can we have $dim (ker(T))<infty$?
    – Kavi Rama Murthy
    Aug 29 at 8:10















up vote
1
down vote

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Let $T:Hto H$ be a continuous, compact, self adjoint linear operator on the Hilbert space H. Can we have $dim ker(T)=infty$?







share|cite|improve this question






















  • Did you intend to ask: "Can we have $dim (ker(T))<infty$?
    – Kavi Rama Murthy
    Aug 29 at 8:10













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 $dim ker(T)=infty$?







share|cite|improve this question














Let $T:Hto H$ be a continuous, compact, self adjoint linear operator on the Hilbert space H. Can we have $dim ker(T)=infty$?









share|cite|improve this question













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edited Aug 29 at 6:37









Arthur

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asked Aug 29 at 6:34









Bogdan

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  • Did you intend to ask: "Can we have $dim (ker(T))<infty$?
    – Kavi Rama Murthy
    Aug 29 at 8:10

















  • Did you intend to ask: "Can we have $dim (ker(T))<infty$?
    – Kavi Rama Murthy
    Aug 29 at 8:10
















Did you intend to ask: "Can we have $dim (ker(T))<infty$?
– Kavi Rama Murthy
Aug 29 at 8:10





Did you intend to ask: "Can we have $dim (ker(T))<infty$?
– Kavi Rama Murthy
Aug 29 at 8:10











1 Answer
1






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up vote
4
down vote



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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: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.






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  • Can you also give a non-trivial example?
    – uniquesolution
    Aug 29 at 7:11










  • @uniquesolution sure
    – Arthur
    Aug 29 at 7:14











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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: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.






share|cite|improve this answer






















  • Can you also give a non-trivial example?
    – uniquesolution
    Aug 29 at 7:11










  • @uniquesolution sure
    – Arthur
    Aug 29 at 7:14















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: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.






share|cite|improve this answer






















  • Can you also give a non-trivial example?
    – uniquesolution
    Aug 29 at 7:11










  • @uniquesolution sure
    – Arthur
    Aug 29 at 7:14













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: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.






share|cite|improve this answer














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: $T(x_1, x_2,ldots) = (x_1, 0,0,0,ldots)$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 29 at 7:15

























answered Aug 29 at 6:40









Arthur

101k795176




101k795176











  • Can you also give a non-trivial example?
    – uniquesolution
    Aug 29 at 7:11










  • @uniquesolution sure
    – Arthur
    Aug 29 at 7:14

















  • Can you also give a non-trivial example?
    – uniquesolution
    Aug 29 at 7:11










  • @uniquesolution sure
    – Arthur
    Aug 29 at 7:14
















Can you also give a non-trivial example?
– uniquesolution
Aug 29 at 7:11




Can you also give a non-trivial example?
– uniquesolution
Aug 29 at 7:11












@uniquesolution sure
– Arthur
Aug 29 at 7:14





@uniquesolution sure
– Arthur
Aug 29 at 7:14


















 

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