An estimate for narrow operators on L p ([0,1])
aa r X i v : . [ m a t h . F A ] A ug AN ESTIMATE FOR NARROW OPERATORS ON L p ([0 , EUGENE SHARGORODSKY and TEO SHARIA
Abstract.
We prove a theorem, which generalises C. Franchetti’s estimatefor the norm of a projection onto a rich subspace of L p ([0 , L p ([0 , ≤ p < ∞ . Introduction
For Banach spaces X and Y , let B ( X, Y ) and K ( X, Y ) denote the sets ofbounded linear and compact linear operators from X to Y , respectively; B ( X ) := B ( X, X ), K ( X ) := K ( X, X ); I ∈ B ( X ) denotes the identity operator. An op-erator P ∈ B ( X ) is called a projection if P = P . A closed linear subspace X ⊂ X is called (in X ) if there exists a projection P ∈ B ( X )such that P ( X ) = X and k P k = 1.Let (Ω , Σ , µ ) be a nonatomic measure space with 0 < µ (Ω) < ∞ . We will usethe following notation: • Σ + := { A ∈ Σ : µ ( A ) > } , • I A is the indicator function of A ∈ Σ, i.e. I A ( ω ) = 1 if ω ∈ A and I A ( ω ) = 0 if ω A , • := I Ω , • E f := (cid:16) µ (Ω) R Ω f dµ (cid:17) .We will use the terminology from [6]. A Σ-measurable function g is called a sign if it takes values in the set {− , , } , and a sign on A ∈ Σ if it is asign with the support equal to A , i.e. if g = I A . A sign is of mean zero if R Ω g dµ = 0.An operator T ∈ B ( L p ( µ ) , Y ), 1 ≤ p < ∞ is called narrow if for every A ∈ Σ + and every ε >
0, there exists a mean zero sign g on A such that k T g k < ε .Every T ∈ K ( L p ( µ ) , Y ) is narrow (see [6, Proposition 2.1]), but there arenoncompact narrow operators. Indeed, let G be a sub- σ -algebra of Σ suchthat there exists a random variable ξ on (cid:16) Ω , Σ , µ (Ω) µ (cid:17) , which is indepen-dent of G and has a nontrivial Gaussian distribution. Then the correspondingconditional expectation operator E G = E ( ·|G ) ∈ B ( L p ( µ )) is narrow (see [6,Corollary 4.25]), but not compact if G has infinitely many pairwise disjointelements of positive measure. Mathematics Subject Classification.
Let(1) C p := max ≤ α ≤ (cid:0) α p − + (1 − α ) p − (cid:1) p (cid:16) α p − + (1 − α ) p − (cid:17) − p for 1 < p < ∞ , and C := 2.In the following theorems, (Ω , Σ , µ ) = ([0 , , L , λ ), where λ is the standardLebesgue measure on [0 ,
1] and L is the σ -algebra of Lebesgue measurablesubsets of [0 , Theorem 1.1 ([3], [4]) . Let P ∈ B ( L p ([0 , \ { } be a narrow projectionoperator, ≤ p < ∞ . Then (2) k I − P k L p → L p ≥ k I − E k L p → L p = C p . The following theorem was proved in [7], where it was used to show that C p is the optimal constant in the bounded compact approximation property of L p ([0 , P = 0 is a finite-rank projection. Theorem 1.2 ([7]) . Let ≤ p < ∞ , γ ∈ C , and let T ∈ K ( L p ([0 , . Then (3) k I − T k L p → L p + inf k u k Lp =1 k ( γI − T ) u k L p ≥ k I − γ E k L p → L p . In particular, (4) k I − T k L p → L p + inf k u k Lp =1 k ( I − T ) u k L p ≥ C p . The following theorem is the main result of the paper. It generalises bothTheorems 1.1 and 1.2.
Theorem 1.3.
Estimates (3) and (4) hold for all narrow operators T ∈B ( L p ([0 , , ≤ p < ∞ . In the case p = 1, inequality (4) (for all narrow operators) follows from aDaugavet-type result due to A.M. Plichko and M.M. Popov ([5, §
9, Theorem8], see also [6, Corollary 6.4]): k I − T k = 1 + k T k for every narrow operator T ∈ B ( L ([0 , . Indeed, k I − T k L → L + inf k u k L =1 k ( I − T ) u k L ≥ k I − T k L → L + 1 − sup k u k L =1 k T u k L = 1 + k T k L → L + 1 − k T k L → L = 2 = C . N ESTIMATE FOR NARROW OPERATORS ON L p ([0 , Proof of Theorem 1.3
It follows from the definition of a narrow operator that if T ∈ B ( L p ( µ )) isnarrow and S ∈ B ( L p ( µ )), then ST ∈ B ( L p ( µ )) is narrow (see [6, Proposition1.8]). On the other hand, there are S, T ∈ B ( L p ( µ )) such that T is narrowbut T S is not (see [6, Proposition 5.1]). The following lemma shows that thelatter cannot happen if S is a multiplication operator. Lemma 2.1.
Let X = L p ( µ ) , g ∈ L ∞ ( µ ) , and T ∈ B ( X, Y ) be a narrowoperator. Then the operator T gI ∈ B ( X, Y ) is also narrow.Proof. There is nothing to prove if T = 0. Suppose T = 0. Take any A ∈ Σ + and any ε >
0. There exists a simple function g = P Mk =1 a k I B k k g − g k L ∞ < ε k T k ( µ (Ω)) /p . Here M ∈ N , B k ∈ Σ, k = 1 , . . . , M are pairwise disjoint, µ ( B k ) > ∪ Mk =1 B k = Ω, a k ∈ C .Let A k := A ∩ B k . If µ ( A k ) >
0, let h k be a mean zero sign on A k such that k T h k k < ε P Mk =1 | a k | . Let h := X { k : µ ( A k ) > } h k . It is clear that h is a mean zero sign on A and k T gIh k ≤ k
T g h k + k T ( g − g ) h k≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T X { k : µ ( A k ) > } a k h k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + k T kk g − g k L ∞ k h k L p < X { k : µ ( A k ) > } | a k |k T h k k + k T k ε k T k ( µ (Ω)) /p ( µ (Ω)) /p ≤ ε P Mk =1 | a k | X { k : µ ( A k ) > } | a k | + ε ≤ ε ε ε. (cid:3) The above lemma and its proof remain valid if X is a K¨othe F-space on(Ω , Σ , µ ) (see [6, Section 1.3]). Similarly, the following lemma and its proofremain valid if X is a rearrangement-invariant Banach space on ([0 , , L , λ )with absolutely continuous norm. This lemma is a minor modification of [6,Theorem 2.21] and [5, §
8, Proposition 5].
Lemma 2.2.
Let X = L p ([0 , and T ∈ B ( X, Y ) be a narrow operator. Thenthere exists a 1-complemented subspace X of X isometrically isomorphic to X such that ∈ X and the restriction T | X of T to X is a compact operator. EUGENE SHARGORODSKY and TEO SHARIA
Proof.
Take a mean zero sign g on [0 ,
1] and set T := T g I . The operator T is narrow according to Lemma 2.1. The proof of [6, Theorem 2.21] (with T in place of T and with ε ≥ k T g k , ε > k T g k ) shows that there existsa 1-complemented (see the proof of [5, §
8, Proposition 5]) subspace X of X isometrically isomorphic to X such that g ∈ X and the restriction T | X of T to X is a compact operator. Let X := g X . Since g = , the operatorof multiplication g I is an isometric isomorphism of X onto X and of X onto itself. Let P ∈ B ( X ) be a projection onto X such that k P k = 1.Then P := g P g I ∈ B ( X ) is a projection onto X such that k P k = 1.Hence X is 1-complemented (this follows also from [1, Theorem 4], since X is isometrically isomorphic to X = L p ([0 , = g ∈ g X = X and T | X = T | X g I | X is compact. (cid:3) Proof of Theorem 1.3.
Take an arbitrary ε >
0. Let(5) δ := inf k u k Lp =1 k ( γI − T ) u k L p . There exists u ∈ L p ([0 , k u k L p = 1 and k ( γI − T ) u k L p < δ + ǫ .Then there exists an approximation h := P Mk =1 a k I A k of u such that A k , k = 1 , . . . , M , M ∈ N are pairwise disjoint Borel sets of positive measure, ∪ Mk =1 A k = [0 , a k ∈ C , and(6) k γh − T h k L p ([0 , ≤ δ + 2 ε, k h k L p ([0 , = 1 . Partition [0 ,
1] into subintervals I k of length λ ( A k ), k = 1 , . . . , M . Since( A k , L , λ ) is isomorphic (modulo sets of measure 0) to ( I k , L , λ ) (see, e.g.,[2, Theorem 9.2.2 and Corollary 6.6.7]), one can easily derive from Lemma 2.2the existence, for each k , of a 1-complemented subspace X k of L p ( A k ) isometri-cally isomorphic to L p ( I k ) such that I A k ∈ X k and T | X k is a compact operator.Let X := { f ∈ L p ([0 , f | A k ∈ X k , k = 1 , . . . , M } . It is easy to see that X is 1-complemented and isometrically isomorphic to L p ([0 , T := T | X is a compact operator. Let J : L p ([0 , → X be an isometric isomorphism and P ∈ B ( L p ([0 , X such that k P k = 1. Then T := J − P T J ∈ K ( L p ([0 , k I − T k X → L p + inf f ∈ X , k f k Lp =1 k ( γI − T ) f k L p ≥ k P ( I − T ) k X → X + inf f ∈ X , k f k Lp =1 k P ( γI − T ) f k L p = k J − P ( I − T ) J k L p → L p + inf ϕ ∈ L p , k ϕ k Lp =1 k J − P ( γI − T ) J ϕ k L p = k I − T k L p → L p + inf ϕ ∈ L p , k ϕ k Lp =1 k ( γI − T ) ϕ k L p ≥ k I − γ E k L p → L p . Since h ∈ X , it follows from (6) that δ + 2 ε ≥ inf f ∈ X , k f k Lp =1 k ( γI − T ) f k L p . N ESTIMATE FOR NARROW OPERATORS ON L p ([0 , Hence k I − T k L p → L p + δ + 2 ε ≥ k I − T k X → L p + inf f ∈ X , k f k Lp =1 k ( γI − T ) f k L p ≥ k I − γ E k L p → L p and k I − T k L p → L p + inf k u k Lp =1 k ( γI − T ) u k L p + 2 ε ≥ k I − γ E k L p → L p for all ε > (cid:3) References [1] T. Ando, Contractive projections in L p -spaces, Pacific J. Math. , 391–405, 1966.[2] V.I. Bogachev, Measure theory. Vol. I and II.
Springer, Berlin, 2007.[3] C. Franchetti, The norm of the minimal projection onto hyperplanes in L p [0 ,
1] and theradial constant,
Boll. Unione Mat. Ital. , VII , Ser., B 4, 4, 803–821, 1990.[4] C. Franchetti, Lower bounds for the norms of projections with small kernels,
Bull. Aust.Math. Soc. , 3, 507–511, 1992.[5] A.M. Plichko and M.M. Popov, Symmetric function spaces on atomless probabilityspaces. Diss. Math. , 85 p., 1990.[6] M. Popov and B. Randrianantoanina,
Narrow operators on function spaces and vectorlattices. de Gruyter Studies in Mathematics , de Gruyter, Berlin, 2013.[7] E. Shargorodsky and T. Sharia, Sharp estimates for conditionally centred moments andfor compact operators on L p spaces, (to appear). E. Shargorodsky, Department of Mathematics, King’s College London, Strand,London WC2R 2LS, United Kingdom and Technische Universit¨at Dresden,Fakult¨at Mathematik, 01062 Dresden, Germany
E-mail address : [email protected] T. Sharia, Department of Mathematics, Royal Holloway, University of Lon-don, Egham, Surrey TW20 0EX, United Kingdom
E-mail address ::