Mixing
Centralizer of projections
Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$,
i.e. $$p^2=p=p^* space text{ and } space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$.
Is it true that $p=pm q$?
Here $C(x)={yin B(H) : yx=xy}$.
Maybe I am asking a well known result but I couldn't find anything about this problem in the literature.
functional-analysis hilbert-spaces operator-algebras compact-operators projection
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
asked Nov 18 '18 at 6:00
golomorfMath
1427
add a comment |
Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$,
i.e. $$p^2=p=p^* space text{ and } space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$.
Is it true that $p=pm q$?
Here $C(x)={yin B(H) : yx=xy}$.
Maybe I am asking a well known result but I couldn't find anything about this problem in the literature.
functional-analysis hilbert-spaces operator-algebras compact-operators projection
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
asked Nov 18 '18 at 6:00
golomorfMath
1427
2
I think you want to say "Is it true that $p = 1 - q$ or $p = q$". Minus a projection is not a projection.
– Adrián González-Pérez
Nov 20 '18 at 12:45
add a comment |
Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$,
i.e. $$p^2=p=p^* space text{ and } space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$.
Is it true that $p=pm q$?
Here $C(x)={yin B(H) : yx=xy}$.
Maybe I am asking a well known result but I couldn't find anything about this problem in the literature.
functional-analysis hilbert-spaces operator-algebras compact-operators projection
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
asked Nov 18 '18 at 6:00
golomorfMath
1427
Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$,
i.e. $$p^2=p=p^* space text{ and } space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$.
Is it true that $p=pm q$?
Here $C(x)={yin B(H) : yx=xy}$.
Maybe I am asking a well known result but I couldn't find anything about this problem in the literature.
functional-analysis hilbert-spaces operator-algebras compact-operators projection
functional-analysis hilbert-spaces operator-algebras compact-operators projection
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
asked Nov 18 '18 at 6:00
golomorfMath
1427
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
asked Nov 18 '18 at 6:00
golomorfMath
1427
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
edited Nov 18 '18 at 8:05
Gaby Alfonso
682315
682315
asked Nov 18 '18 at 6:00
golomorfMath
1427
asked Nov 18 '18 at 6:00
golomorfMath
1427
asked Nov 18 '18 at 6:00
golomorfMath
1427
1427
2
I think you want to say "Is it true that $p = 1 - q$ or $p = q$". Minus a projection is not a projection.
– Adrián González-Pérez
Nov 20 '18 at 12:45
add a comment |
2
I think you want to say "Is it true that $p = 1 - q$ or $p = q$". Minus a projection is not a projection.
– Adrián González-Pérez
Nov 20 '18 at 12:45
2
2
I think you want to say "Is it true that $p = 1 - q$ or $p = q$". Minus a projection is not a projection.
– Adrián González-Pérez
Nov 20 '18 at 12:45
I think you want to say "Is it true that $p = 1 - q$ or $p = q$". Minus a projection is not a projection.
– Adrián González-Pérez
Nov 20 '18 at 12:45
add a comment |
1 Answer
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Look in Vaughan Jones notes, Exerise 2.1.13. $A$ and $A^ast$ preserve a subspace $K$, ie $A K subset K$ and $A^ast K subset K$ iff $[A,P_K] = 0$, where $P_K$ is the orthogonal projection onto $K$.
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
add a comment |
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1 Answer
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Look in Vaughan Jones notes, Exerise 2.1.13. $A$ and $A^ast$ preserve a subspace $K$, ie $A K subset K$ and $A^ast K subset K$ iff $[A,P_K] = 0$, where $P_K$ is the orthogonal projection onto $K$.
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
add a comment |
Look in Vaughan Jones notes, Exerise 2.1.13. $A$ and $A^ast$ preserve a subspace $K$, ie $A K subset K$ and $A^ast K subset K$ iff $[A,P_K] = 0$, where $P_K$ is the orthogonal projection onto $K$.
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
add a comment |
Look in Vaughan Jones notes, Exerise 2.1.13. $A$ and $A^ast$ preserve a subspace $K$, ie $A K subset K$ and $A^ast K subset K$ iff $[A,P_K] = 0$, where $P_K$ is the orthogonal projection onto $K$.
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
Look in Vaughan Jones notes, Exerise 2.1.13. $A$ and $A^ast$ preserve a subspace $K$, ie $A K subset K$ and $A^ast K subset K$ iff $[A,P_K] = 0$, where $P_K$ is the orthogonal projection onto $K$.
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
answered Nov 20 '18 at 12:46
Adrián González-Pérez
971138
971138
add a comment |
add a comment |
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2
I think you want to say "Is it true that $p = 1 - q$ or $p = q$". Minus a projection is not a projection.
– Adrián González-Pérez
Nov 20 '18 at 12:45