Ideal of strictly singular operatorsStrictly singular operators and their adjointsExistence of injective operators with dense rangeOperators from $ell_infty$Non strictly-singular operators and complemented subspacesUnconditionally $p$-converging operators on $L_1[0,1]$Self-adjoint, strictly singular operators on Hilbert spacesExample of a strictly cosingular operator whose dual is not strictly singular?Operational quantities characterizing strictly singular operators and strictly cosingular operatorsA question on operational quantities characterizing strictly singular operatorsThe dual relationships between strictly singular operators and strictly cosingular operators
Ideal of strictly singular operators
Strictly singular operators and their adjointsExistence of injective operators with dense rangeOperators from $ell_infty$Non strictly-singular operators and complemented subspacesUnconditionally $p$-converging operators on $L_1[0,1]$Self-adjoint, strictly singular operators on Hilbert spacesExample of a strictly cosingular operator whose dual is not strictly singular?Operational quantities characterizing strictly singular operators and strictly cosingular operatorsA question on operational quantities characterizing strictly singular operatorsThe dual relationships between strictly singular operators and strictly cosingular operators
$begingroup$
Let $X$ be a Banach space. An operator $T:Xto X$ is called strictly singular iff for any infinite dimensional subspace $Ysubseteq X,$ $T|_Y:Yto T(Y)$ is not an isomorphism.
It is known that for $X=ell_p,$ $1leq p<infty,$ an operator is strictly singular iff it is compact. Also $T:ell_inftytoell_infty$ is strictly singular iff $T$ is weakly compact. Can someone provide me proofs for these facts? I could not really locate proofs of the above mentioned facts in literature.
fa.functional-analysis banach-spaces operator-theory
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Banach space. An operator $T:Xto X$ is called strictly singular iff for any infinite dimensional subspace $Ysubseteq X,$ $T|_Y:Yto T(Y)$ is not an isomorphism.
It is known that for $X=ell_p,$ $1leq p<infty,$ an operator is strictly singular iff it is compact. Also $T:ell_inftytoell_infty$ is strictly singular iff $T$ is weakly compact. Can someone provide me proofs for these facts? I could not really locate proofs of the above mentioned facts in literature.
fa.functional-analysis banach-spaces operator-theory
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Banach space. An operator $T:Xto X$ is called strictly singular iff for any infinite dimensional subspace $Ysubseteq X,$ $T|_Y:Yto T(Y)$ is not an isomorphism.
It is known that for $X=ell_p,$ $1leq p<infty,$ an operator is strictly singular iff it is compact. Also $T:ell_inftytoell_infty$ is strictly singular iff $T$ is weakly compact. Can someone provide me proofs for these facts? I could not really locate proofs of the above mentioned facts in literature.
fa.functional-analysis banach-spaces operator-theory
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
$endgroup$
Let $X$ be a Banach space. An operator $T:Xto X$ is called strictly singular iff for any infinite dimensional subspace $Ysubseteq X,$ $T|_Y:Yto T(Y)$ is not an isomorphism.
It is known that for $X=ell_p,$ $1leq p<infty,$ an operator is strictly singular iff it is compact. Also $T:ell_inftytoell_infty$ is strictly singular iff $T$ is weakly compact. Can someone provide me proofs for these facts? I could not really locate proofs of the above mentioned facts in literature.
fa.functional-analysis banach-spaces operator-theory
fa.functional-analysis banach-spaces operator-theory
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
edited Jun 18 at 7:36
Yemon Choi
16.9k5 gold badges49 silver badges107 bronze badges
16.9k5 gold badges49 silver badges107 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
asked Jun 18 at 7:13
Samya Kumar RaySamya Kumar Ray
1315 bronze badges
1315 bronze badges
add a comment |
add a comment |
1 Answer
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[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $T:ell_ptoell_p$ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $T:C(K)to X$ is strictly singular, and Theorem 5.5.3 says that a non-weakly compact operator $T:C(K)to X$ is not strictly singular.
Note that $ell_infty$ is a $C(K)$ space with $K$ the Stone-Cech compactification of the set of positive integers.
answered Jun 18 at 8:13
M.GonzálezM.González
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1 Answer
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$begingroup$
[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $T:ell_ptoell_p$ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $T:C(K)to X$ is strictly singular, and Theorem 5.5.3 says that a non-weakly compact operator $T:C(K)to X$ is not strictly singular.
Note that $ell_infty$ is a $C(K)$ space with $K$ the Stone-Cech compactification of the set of positive integers.
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
$endgroup$
add a comment |
$begingroup$
[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $T:ell_ptoell_p$ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $T:C(K)to X$ is strictly singular, and Theorem 5.5.3 says that a non-weakly compact operator $T:C(K)to X$ is not strictly singular.
Note that $ell_infty$ is a $C(K)$ space with $K$ the Stone-Cech compactification of the set of positive integers.
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
$endgroup$
add a comment |
$begingroup$
[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $T:ell_ptoell_p$ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $T:C(K)to X$ is strictly singular, and Theorem 5.5.3 says that a non-weakly compact operator $T:C(K)to X$ is not strictly singular.
Note that $ell_infty$ is a $C(K)$ space with $K$ the Stone-Cech compactification of the set of positive integers.
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
$endgroup$
[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $T:ell_ptoell_p$ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $T:C(K)to X$ is strictly singular, and Theorem 5.5.3 says that a non-weakly compact operator $T:C(K)to X$ is not strictly singular.
Note that $ell_infty$ is a $C(K)$ space with $K$ the Stone-Cech compactification of the set of positive integers.
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
edited Jun 18 at 14:27
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
answered Jun 18 at 8:13
M.GonzálezM.González
1,9688 silver badges16 bronze badges
1,9688 silver badges16 bronze badges
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