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Frames for linear operators in Banach spaces
Oleg Reinov
Oleg Reinov Frames for linear operators in Banach spaces
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Schauder frames
Recall only a definition of a Schauder frame for a Banach space: asystem (x ′k)∞k=1 ⊂ X ∗, (xk)∞k=1 ⊂ X is said to be a Schauder framefor X , if for every x ∈ X the series
∑∞k=1〈x ′k , x〉xk converges in X
to x .To find information and History:
P.G. Casazza, S.J. Dilworth, E. Odell, Th. Schlumprecht, A.Zsak: Coefficient quantization for frames in Banach spaces, J.Math. Anal. Appl., 348 (2008), 66-86.
D. Han, D.H. Larson: Frames, bases, and group representation,Mem. Amer. Math. Soc., 147 (2000), x+94 pp.
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Definition
A system F := ((x ′k)∞k=1, (yk)∞k=1) is an O-frame (operator frame)for T ∈ L(X ,Y ), if for every x ∈ X the series
∑∞k=1〈x ′k , x〉yk
converges in Y and
Tx =∞∑k=1
〈x ′k , x〉yk , x ∈ X .
Examples. 1. ∆ : l∞ → l1, a diagonal, (δk) ∈ l1. Then∆x =
∑δk 〈ek , x〉ek .
2. X has a basis (fk)∞k=1 If T : X →W , then
Tx =∞∑k=1
〈f ′k , x〉Tfk , x ∈ X .
3. W has a basis (wk). If T : X →W , then〈Tx ,w ′k〉 = 〈x ,T ∗w ′k〉, hence Tx =
∑∞k=1〈T ∗w ′k , x〉wk .
4. If X (or Y ) is separable and has BAP. Every T ∈ L(X ,W ) hasO-frame.
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Definition
A system F := ((x ′k)∞k=1, (yk)∞k=1) is an O-frame (operator frame)for T ∈ L(X ,Y ), if for every x ∈ X the series
∑∞k=1〈x ′k , x〉yk
converges in Y and
Tx =∞∑k=1
〈x ′k , x〉yk , x ∈ X .
Examples. 1. ∆ : l∞ → l1, a diagonal, (δk) ∈ l1. Then∆x =
∑δk 〈ek , x〉ek .
2. X has a basis (fk)∞k=1 If T : X →W , then
Tx =∞∑k=1
〈f ′k , x〉Tfk , x ∈ X .
3. W has a basis (wk). If T : X →W , then〈Tx ,w ′k〉 = 〈x ,T ∗w ′k〉, hence Tx =
∑∞k=1〈T ∗w ′k , x〉wk .
4. If X (or Y ) is separable and has BAP. Every T ∈ L(X ,W ) hasO-frame.
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Ideal property
Proposition
T ∈ L(X ,W ), A ∈ L(W ,V ),B ∈ L(Z ,X ). If T has an O-frame,then ATB : Z → V has an O-frame.
Corollary
If T ∈ L(X ,W ) factors through a Banach space with a basis, thenT has an O-frame.
Corollary
If T ∈ L(X ,W ) factors through a Banach space with the BAP,then T has an O-frame.
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Ideal property
Proposition
T ∈ L(X ,W ), A ∈ L(W ,V ),B ∈ L(Z ,X ). If T has an O-frame,then ATB : Z → V has an O-frame.
Corollary
If T ∈ L(X ,W ) factors through a Banach space with a basis, thenT has an O-frame.
Corollary
If T ∈ L(X ,W ) factors through a Banach space with the BAP,then T has an O-frame.
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Ideal property
Proposition
T ∈ L(X ,W ), A ∈ L(W ,V ),B ∈ L(Z ,X ). If T has an O-frame,then ATB : Z → V has an O-frame.
Corollary
If T ∈ L(X ,W ) factors through a Banach space with a basis, thenT has an O-frame.
Corollary
If T ∈ L(X ,W ) factors through a Banach space with the BAP,then T has an O-frame.
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Dual situation
One more property:
Proposition
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). Then thedual system Fd := ((wk), (x ′k)) is a weak∗ O-frame for T ∗, i.e.
T ∗w ′ = w∗- limN
N∑k=1
〈w ′,wk〉x ′k , w ′ ∈W ∗.
Proof. For w ′ ∈W ∗ and x ∈ X we have:
〈Tx ,w ′〉 = 〈∞∑k=1
〈x ′k , x〉wk ,w′〉 = 〈
∞∑k=1
〈w ′,wk〉x ′k , x〉,
hence T ∗w ′ = w∗- limN∑N
k=1 x′k〈w ′,wk〉 (the limit is in the
topology σ(X ∗,X )).
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Dual situation
One more property:
Proposition
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). Then thedual system Fd := ((wk), (x ′k)) is a weak∗ O-frame for T ∗, i.e.
T ∗w ′ = w∗- limN
N∑k=1
〈w ′,wk〉x ′k , w ′ ∈W ∗.
Proof. For w ′ ∈W ∗ and x ∈ X we have:
〈Tx ,w ′〉 = 〈∞∑k=1
〈x ′k , x〉wk ,w′〉 = 〈
∞∑k=1
〈w ′,wk〉x ′k , x〉,
hence T ∗w ′ = w∗- limN∑N
k=1 x′k〈w ′,wk〉 (the limit is in the
topology σ(X ∗,X )).
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Dual O-frame
Definition
O-frame ((x ′k), (wk)) for T is shrinking if for every w ′ ∈W ∗ thenorm ||
∑∞k=n+1 x
′k〈w ′,wk〉|| → 0 as n→∞.
Proposition
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). The dualsystem Fd := ((wk), (x ′k)) is an O-frame for T ∗ iff the O-frame Fis shrinking.
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Dual O-frame
Definition
O-frame ((x ′k), (wk)) for T is shrinking if for every w ′ ∈W ∗ thenorm ||
∑∞k=n+1 x
′k〈w ′,wk〉|| → 0 as n→∞.
Proposition
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). The dualsystem Fd := ((wk), (x ′k)) is an O-frame for T ∗ iff the O-frame Fis shrinking.
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Boundedly complete O-frame
Definition
O-frame ((x ′k), (wk)) for T is boundedly complete if for everyx ′′ ∈ X ∗∗ the series
∑∞k=1〈x ′′, x ′k〉wk converges in the space W .
Proposition
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). TFAE:1) O-frame F is boundedly complete;2) for every x ′′ ∈ X ∗∗, it follows from the boudedness of thepartial sums (
∑Nk=1〈x ′′, x ′k〉wk)∞N=1 the convergence of the series∑∞
k=1〈x ′′, x ′k〉wk in the space W .
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Boundedly complete O-frame
Definition
O-frame ((x ′k), (wk)) for T is boundedly complete if for everyx ′′ ∈ X ∗∗ the series
∑∞k=1〈x ′′, x ′k〉wk converges in the space W .
Proposition
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). TFAE:1) O-frame F is boundedly complete;2) for every x ′′ ∈ X ∗∗, it follows from the boudedness of thepartial sums (
∑Nk=1〈x ′′, x ′k〉wk)∞N=1 the convergence of the series∑∞
k=1〈x ′′, x ′k〉wk in the space W .
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O-frames and weak compactness
Theorem
Let F := ((x ′k), (wk)) be an O-frame for T ∈ L(X .W ). If thisO-frame F is boundedly complete and shrinking, then the operatorT is weakly compact.
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O-frames and basis-factorization
Theorem
Let T ∈ L(X ,W ). TFAE:1) T has an O-frame;2) the operator T factors through a Banach space with a basis;3) T factors through a Banach sequence space with a basis.
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O-frames and basis-factorization: Proof
Proof.
T , O-frame F := ((x ′k), (wk)), wk 6= 0.
∃K > 0 : ∀N ||∑N
k=1 x′k ⊗ wk || ≤ K .
t := {a = (ak)∞k=1 : series∑∞
k=1 akwk converges in W },|||a|||t := supN ||
∑Nk=1 akwk || (≥ limN ||
∑Nk=1 akwk ||). For
a = (a1, a2, . . . , aN+s , 0, 0, . . . ), |||∑N
k=1 akek ||| ≤ |||∑N+s
k=1 akek |||and the linear span of (ek)∞k=1 is dense in t. Thus, (ek) is amonotonr basis in the Banach space t. If j : t →W is a naturalmap a 7→
∑∞k=1 akwk , then ||j || ≤ 1. Set Ax := (〈x ′k , x〉)∞k=1; then
Ax ∈ t. Furthermore,
|||Ax |||t = supN||
N∑k=1
〈x ′k , x〉wk || ≤ K ||x ||, ∀ x ∈ X .
Thus, A ∈ L(X , t) and T = jA : X → t →W .
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Unconditional O-frames
Definition
Let T ∈ L(X ,W ), (x ′k)∞k=1 ⊂ X ∗, (wk)∞k=1 ⊂W . We say thatF := ((x ′k)∞k=1, (wk)∞k=1) is an UO-frame (unconditional operatorframe) for T , if for every x ∈ X the series
∑∞k=1〈x ′k , x〉wk
converges unconditionally in W and
Tx =∞∑k=1
〈x ′k , x〉wk , x ∈ X .
Theorem
Let T ∈ L(X ,W ). TFAE:1) T has a UO-frame;2) T the operator T factors through a Banach space with anunconditional basis;3) T the operator T factors through a Banach sequence spacewith an unconditional basis basis.
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Unconditional O-frames
Definition
Let T ∈ L(X ,W ), (x ′k)∞k=1 ⊂ X ∗, (wk)∞k=1 ⊂W . We say thatF := ((x ′k)∞k=1, (wk)∞k=1) is an UO-frame (unconditional operatorframe) for T , if for every x ∈ X the series
∑∞k=1〈x ′k , x〉wk
converges unconditionally in W and
Tx =∞∑k=1
〈x ′k , x〉wk , x ∈ X .
Theorem
Let T ∈ L(X ,W ). TFAE:1) T has a UO-frame;2) T the operator T factors through a Banach space with anunconditional basis;3) T the operator T factors through a Banach sequence spacewith an unconditional basis basis.
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O-frames and bounded approximation property
Definition
Let T ∈ L(X ,W ), C ≥ 1. We say that T has the C-BAP if forevery compact subset K of X , for every ε > 0 there is a finite rankoperator R : X →W such that ||R|| ≤ C ||T || andsupx∈K ||Rx − Tx || ≤ ε. T has the BAP, if it has the C-BAP forsome C ∈ [1,∞).
Theorem
Let X be a separable Banach space, W be any Banach space andT ∈ L(X ,W ). TFAE:(1) T has an O-frame;(2) T has the BAP;(3) T factors through a Banach space with a basis.
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O-frames and bounded approximation property
Definition
Let T ∈ L(X ,W ), C ≥ 1. We say that T has the C-BAP if forevery compact subset K of X , for every ε > 0 there is a finite rankoperator R : X →W such that ||R|| ≤ C ||T || andsupx∈K ||Rx − Tx || ≤ ε. T has the BAP, if it has the C-BAP forsome C ∈ [1,∞).
Theorem
Let X be a separable Banach space, W be any Banach space andT ∈ L(X ,W ). TFAE:(1) T has an O-frame;(2) T has the BAP;(3) T factors through a Banach space with a basis.
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Comparing Banach frames and O-frames
Comparing the usual Banach frames with O-frames.
Known: If X has an unconditional Banach frame, then:1. The frame is shrinking iff X does not contain l1 iff X isalmost reflexive.2. X is reflexive iff it does not contain both l1 and c0.
For O-frames, the situation is different. We have
Example
There exists an operator T : l1 → C [0, 1] such that1. T is conditionally weakly compact and, thus, does not containl1. T has no shrinking O-frame.2. T does not contain also c0, but is not weakly compact.
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Comparing Banach frames and O-frames
Comparing the usual Banach frames with O-frames.
Known: If X has an unconditional Banach frame, then:1. The frame is shrinking iff X does not contain l1 iff X isalmost reflexive.2. X is reflexive iff it does not contain both l1 and c0.
For O-frames, the situation is different. We have
Example
There exists an operator T : l1 → C [0, 1] such that1. T is conditionally weakly compact and, thus, does not containl1. T has no shrinking O-frame.2. T does not contain also c0, but is not weakly compact.
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Comparing Banach frames and O-frames
Trivially, If X does not have the approximation property, thenit can not have a Banach frame.For O-frames, the situation is different. We have
Example
There exist two separable reflexive Banach spaces X ,Y and and anoperator T : X → Y so that:Both X and Y do not have the approximation property, but T hasan unconditional O-frame.
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Comparing Banach frames and O-frames
Trivially, If X does not have the approximation property, thenit can not have a Banach frame.For O-frames, the situation is different. We have
Example
There exist two separable reflexive Banach spaces X ,Y and and anoperator T : X → Y so that:Both X and Y do not have the approximation property, but T hasan unconditional O-frame.
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Comparing Banach frames and O-frames
Known: If a reflexive space has the approximation property,then it has a Banach frame.
For O-frames, the situation is different. We have
Example
There exists a weakly compact operator, which has theapproximation property, but has no operator frame.
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Comparing Banach frames and O-frames
Known: If a reflexive space has the approximation property,then it has a Banach frame.
For O-frames, the situation is different. We have
Example
There exists a weakly compact operator, which has theapproximation property, but has no operator frame.
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Thank you for your attention!
Oleg Reinov Frames for linear operators in Banach spaces