aa r X i v : . [ m a t h . F A ] D ec On operators with bounded approximation property
Oleg Reinov
Abstract.
It is known that any separable Banach space with BAP is a comple-mented subspace of a Banach space with a basis. We show that every operatorwith bounded approximation property, acting from a separable Banach space, canbe factored through a Banach space with a basis. §1. LemmasDefinition 1.1.
Let T ∈ L ( X, W ) , C ≥ . We say that T has the C-BAP(C-bounded approximation property) if for every compact subset K of X, for any ε > there exists a finite rank operator R : X → W such that || R || ≤ C || T || and sup x ∈ K || Rx − T x || ≤ ε. The operator T has the BAP if it has the C-BAP for some C ∈ [1 , ∞ ) . Lemma 1.1. T has C-BAP iff for any finite family ( x k ) Nk =1 ⊂ X, for any ε > there exists a finite rank operator R : X → W such that || R || ≤ C || T || and sup ≤ k ≤ N || Rx k − T x k || ≤ ε. Proof . We may (and do) assume that || T || = 1 . Fix a compact subset K ⊂ X and ε > . Let ε := ε/ (2 + C ) , ( x k ) Mk =1 be an ε -net for K in X, R ∈ X ∗ ⊗ W, || R || ≤ C and sup ≤ k ≤ M || Rx k − T x k || ≤ ε . Take an x ∈ K, and let x k be suchthat || x − x k || ≤ ε . Then || T x − Rx || ≤ || x − x k || + || T x k − Rx k || + || Rx k − Rx || ≤ ε + ε + Cε = ε. Lemma 1.2.
Let
X, W be Banach spaces, X being separable, and T ∈ L ( X, W ) .T has C-BAP iff there exists a sequence ( Q l ) ∞ l =1 of finite rank operators from X to W such that for every x ∈ X the series P ∞ l =1 Q l x converges and T x = ∞ X l =1 Q l x, x ∈ X ;2) sup N || P Nl =1 Q l || ≤ C || T || . Proof . Since X is separable, there exists a sequence ( x k ) ∞ which is dense in theclosed unit ball ¯ B (0) of X. Suppose as above that || T || = 1 and T has the C-BAP,that is for any finite set F ⊂ X, for every ε > there is a finite rank operator R : X → W such that || R || ≤ C and sup f ∈ F || Rf − T f || ≤ ε. Put, for N = 1 , , . . . , AMS Subject Classification 2010: 46B28 Spaces of operators; tensor products; approximationproperties.Key words: 46B28 Approximation of operators; bounded approximation property.
1N OPERATORS WITH BOUNDED APPROXIMATION PROPERTY 2 F N := span( x k ) Nk =1 ; F N ⊂ F N +1 . . . . For each N, let R N be a finite rank operatorfrom X to W with the properties that(i) || R N || ≤ C and(ii) sup ≤ n ≤ N || R N x n − T x n || ≤ / N +1 . If n ∈ N then for every N ≥ n one has ( iii ) || R N x n − T x n || ≤ N +1 and, therefore, for a fixed x n R N x n → T x n as N tends to ∞ . Now, fix ε > and let δ > be such that Cδ + δ < ε. For x ∈ ¯ B (0) , take an x n with || x n − x || < δ. Then there is an N so that for N ≥ N || R N x − T x || ≤ || R N || || x n − x || + || R N x n − T x n || + || T x n − T x || ≤ Cδ + || T || δ < ε. Thus, if x ∈ X then R N x → T x as N → + ∞ . To finish the proof of the "only if" part, we apply
Lemma 1.3.
Let
X, W be any Banach spaces, C ≥ and T ∈ L ( X, W ) . Suppose that(*) there exists a sequence ( S N ) ∞ N =1 of finite rank operators from X to W suchthat if x ∈ X then S N x → T x as N → + ∞ and || S N || ≤ C || T || for every N. Then there exists a sequence ( Q l ) ∞ l =1 of finite rank operators from X to W suchthat for every x ∈ X the series P ∞ l =1 Q l x converges and T x = ∞ X l =1 Q l x, x ∈ X ;2) sup N || P Nl =1 Q l || ≤ C || T || . Proof . We assume again that || T || = 1 . Put Q := S , Q l := S l − S l − for l > , so that S N = S + ( S − S ) + · · · + ( S N − − S N − ) + ( S N − S N − ) = Q + Q + · · · + Q N . It follows that (1)
T x = ∞ X l =1 Q l x ∀ x ∈ X and (2) sup N || N X l =1 Q j || = sup N || S N || ≤ C. The "if" part of the proof of Lemma 1.2 follows from
Lemma 1.4.
Let
X, W be any Banach spaces, C ≥ and T ∈ L ( X, W ) . If thereexists a sequence ( R N ) ∞ of finite rank operators from X into W which convergespointwise to T and such that || R N || ≤ C || T || for all N then T has the C-BAP. N OPERATORS WITH BOUNDED APPROXIMATION PROPERTY 3
Proof . Indeed, let R N x → T x for every x ∈ X (and || R N || ≤ C || T || ). Fix ε > and a compact subset K ⊂ X. Put ε := ε ( || T || + 1 + C || T || ) − . Take a finite ε -net F ⊂ X for K and consider R N such that sup f ∈ F || R N f − T f || ≤ ε . Then, for any x ∈ K there is an f ∈ F with || f − x || ≤ ε , and one has : || T x − R N x || ≤ || T || ε + ε + || R N || ε ≤ ε ( || T || + 1 + C || T || ) = ε. Corollary 1.1. If X is separable and T ∈ L ( X, W ) , then T has the C-BAPiff there exists a sequence ( R N ) ∞ of finite rank operators from X into W whichconverges pointwise to T and such that || R N || ≤ C || T || for all N. Corollary 1.2. If X is separable and T ∈ L ( X, W ) , then T has the BAPiff there exists a sequence of finite rank operators from X to W convergent to T pointwise. §2. Theorem. Now, we redenote some objects from §1. Let
X, W be any Banach spaces, T ∈ L ( X, W ) and T possesses the property (*) from Lemma 1.3. Consider the sequence ( Q l ) ∞ l =1 , given by assertion of Lemma 1.3, and put A p := Q p ( p = 1 , , . . . ) and K := C ( ≥ , assuming that || T || = 1 . We are now in notations (partially) of thepaper [1].
Theorem 2.1. If T : X → W has the property (*), then there exist a Banachspace Y with a Schauder basis and two operators e A : X → Y and j : Y → W sothat T = j e A. Proof . In the above notation (assuming that || T || = 1 ), we have: T x = ∞ X p =1 A p x, ∀ x ∈ X ; A p ∈ X ∗ ⊗ W, sup n ∈ N || n X p =1 A p || ≤ K (note that for every n || A n || ≤ || P np =1 A p − P n − p =1 A p || ≤ K ). Let E p = A p ( X ) ⊂ W, m p := dim E p for p ≥ and m = 0 . We will proceed as in [1].By Auerbach, there exist one-dimensional operators B ( p ) j : E p → E p with || B ( p ) j || =1 for j = 1 , , . . . , m p , and so that m p X j =1 B ( p ) j ( e ) = e, e ∈ E p . Set C ( p ) i := 1 m p B ( p ) j for i = rm p + j (where r = 0 , , . . . , m p − j = 1 , , . . . , m p ) . Then, for e ∈ E p . m p X i =1 C ( p ) i ( e ) = m p · m p X j =1 m p B ( p ) j e = m p X j =1 B ( p ) j e = e. N OPERATORS WITH BOUNDED APPROXIMATION PROPERTY 4
Also, for any q ≥ , q ≤ m p and some l < m p and k ≤ m p we have: || q X i =1 C ( p ) i || = || lm p X i =1 C ( p ) i + lm p + k X lm p +1 C ( p ) i || ≤ l · m p || m p X j =1 B ( p ) j || + 1 m p ·|| k X j =1 B ( p ) j || ≤ . Now, let e A s := C ( p ) i A p for p ∈ N , i = 1 , , . . . , m p and s = m + m + · · · + m p − + i. e A s maps X into E p ⊂ W in the following way: e A s : X A p → E p = A p ( X ) C ( p ) i → E p ( ⊂ W ) . Since, for any n ∈ N , for some k and r ≤ m kn X s =1 e A s = k − X p =1 m p X i =1 C ( p ) i A p + r X i =1 C ( k ) i A k , we get that || n X s =1 e A s || ≤ || k − X p =1 A p || + || r X i =1 C ( k ) i A k || ≤ K + 2 || A k || ≤ K. (To get an estimation " K " as in [1], it it enough to consider simultaniousely, in thecenter, the given sum and the sum like || P kp =1 A p || + || P m k i = r C ( k ) i A k || ).Since, for every x ∈ X, A k x → as k → ∞ , we have: lim n →∞ n X s =1 e A s x = lim N →∞ N X p =1 A p x = T x.
Now, consider the space Y := { ( y ( s )) ∞ s =1 : y ( s ) ∈ e A s ( X ) , ∞ X s =1 y ( s ) converges in W } . Set ||| ( y ( s )) ∞ ||| := sup n || P ns =1 y ( s ) || W . Note that ( e A s ( x )) ∞ s − ∈ Y for every x ∈ X, P ∞ s − e A s ( x ) = T x and ||| ( e A s ( x )) ∞ s =1 ||| Y ≤ K || x || X . Therefore, the map e A : X → Y, defined by e A ( x ) = ( e A s ( s )) ∞ s =1 , is linear and continuous (and || e A || ≤ K ). Let j : Y → W be the natural map which takes ( y ( s )) ∞ s =1 to P ∞ s =1 y ( s ) . Since || ∞ X s =1 y ( s ) || W = lim N || N X s =1 y ( s ) || W ≤ sup n || n X s =1 y ( s ) || W , then || j || L ( Y,W ) ≤ . Therefore,
T x = j e A : X → Y → W. It remains now to considerthe space Y. For each s, let e y s ∈ e A s be of norm 1. If ( y s ) ∞ s =1 ∈ Y, then y s = c s e y s . Define ¯ y s ∈ Y by ¯ y s ( t ) = 0 for t = s and ¯ y s ( s ) = e y s ( s = 1 , , . . . ) . Then, every y = ( y s ) ∈ Y is of N OPERATORS WITH BOUNDED APPROXIMATION PROPERTY 5 type P ∞ s =1 c s ¯ y s , if we consider (¯ y s ) ∞ s =1 as a basis in Y. And this basis is monotone:for all scalars ( c s ) we have that ||| m X s =1 c s ¯ y s ||| ≤ ||| m +1 X s =1 c s ¯ y s ||| (by definition of the norm in Y ) . Finally, the space Y is Banach (cf. [2,p. 18, Prop.3.1]). Remark . The theorem just obtained is a spade-theorem for some futher investi-gations in a next paper.
Corollary 2.1. If X is separable and T ∈ L ( X, W ) then T has the boundedapproximation property if and only if T can be factored through a Banach spacewith a basis. References [1] A. Pe l czy´nski : Any separable Banach space with the bounded approximation property is acomplemented subspace of a Banach space with a basis , Studia Math., XL (1971), 239-243.[2] I. Singer: Bases in Banach spaces I , Berlin-Heidelberg-New York: Springer (1970).
St. Petersburg State University, Saint Petersburg, RUSSIA.
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