Contractive projections in Orlicz sequence spaces

We characterize norm one complemented subspaces of Orlicz sequence spaces $\ell_M$ equipped with either Luxemburg or Orlicz norm, provided that the Orlicz function $M$ is sufficiently smooth and sufficiently different from the square function. This paper concentrates on the more difficult real case, the complex case follows from previously known results.


Introduction
One of the main topics in the study of Banach spaces has been, since the inception of the field, the study of projections and complemented subspaces. Naturally, one of the most important topics of the isometric Banach space theory is the study of contractive projections (i.e. projections of norm one) and 1-complemented subspaces (i.e. ranges of norm one projections). They were also investigated from the approximation theory point of view, as part of of the study of minimal projections, i.e. projections onto the given subspace with the smallest possible norm, for an overview of this line of research see [4,12]. Contractive projections are also closely related to the metric projections or nearest point mappings, and are a natural extension of the notion of orthogonal projections from the Hilbert space setting to general Banach spaces. We refer the reader to the survey [16] for an outline of the development and applications of this theory. Here we just indicate some main facts putting the results of the present paper in context.
It is well known that in Lebesgue spaces L p and ℓ p , 1 ≤ p < ∞, a subspace Y is 1complemented if and only if Y is isometrically isomorphic to an L p −space of appropriate dimension (see [1,5]). This is no longer the case for other spaces. Lindberg [8] demonstrated that there exist classes of Orlicz sequence spaces ℓ M containing 1-complemented subspaces which are not even isomorphic to ℓ M . In fact, he showed that for all 1 < a ≤ b < ∞, there exists a reflexive Orlicz sequence space ℓ M so that for all p ∈ [a, b] there is a contractive projection from ℓ M onto a subspace isomorphic to ℓ p . This implies in particular that Orlicz sequence spaces can have continuum isomorphic types of 1-complemented subspaces and thus any attempt for a geometric characterization of 1-complemented subspaces seemed hopeless.
On the other hand, 1-complemented subspacesof ℓ p are also characterized as subspaces which are spanned by a family of mutually disjoint elements of ℓ p (see [9,2]). Moreover all known examples of 1-complemented subspaces in symmetric Banach spaces with 1-unconditional bases, and sufficiently different from Hilbert spaces, are spanned by a family of mutually disjoint vectors. (Note here that, since in Hilbert spaces every subspace is 1-complemented, it is both natural and necessary to include in this context some kind of an assumption about the space being different from Hilbert space.) In particular, the above described example of Lindberg of 1-complemented subspaces of Orlicz sequence spaces which were pathological in the isomorphic sense, are not pathological in the sense that they are spanned by mutually disjoint vectors and the norm one projection is the most natural averaging projection. It was shown in [13] that indeed every 1-complemented subspace Y in any complex Banach space X with a 1-unconditional basis (not necessarily symmetric) which does not contain a 1-complemented isometric copy of a 2-dimensional Hilbert space ℓ 2 2 , has to be spanned by a family of disjointly supported elements of X and the norm one projection from X onto Y has to be the averaging projection. In particular, this holds in complex Orlicz sequence spaces ℓ M equipped with either the Luxemburg or the Orlicz norm when M is sufficiently different from the square function (cf. Remark 4.5).
In the real case this statement in its full generality is false (cf. [13]). For real spaces we only had the following much less satisfactory result describing special 1-complemented subspaces of finite codimension in Orlicz sequence spaces ℓ M . Theorem 1.1. [14,Theorem 7] Let M be an Orlicz function such that M(t) > 0 for all t > 0 and M is not similar to t 2 (i.e. there do not exist constants C, t 0 > 0 so that M(t) = Ct 2 for all t < t 0 ). Let ℓ M be the Orlicz space equipped with either the Luxemburg or the Orlicz norm and F ⊂ ℓ M be a subspace of finite codimension. If F contains at least one basis vector and F is 1-complemented in ℓ M then F is spanned by a family of disjointly supported vectors.
In the present paper we prove a much stronger result -we eliminate the assumption that the subspace should be of finite codimension. Namely we show that when M is a sufficiently smooth Orlicz function which satisfies condition ∆ 2 and is sufficiently different from the square function, then every 1-complemented subspace of the real Orlicz space ℓ M is spanned by a family of mutually disjoint vectors and every norm one projection in ℓ M is an averaging projection (see Theorem 4.3 and Corollary 4.4). This result is valid in Orlicz spaces equipped with either the Luxemburg or the Orlicz norm.
Our method of proof is different from that of [14], it relies on new results characterizing averaging projections through properties related to and generalizing disjointness preserving operators [17].
Recently, Jamison, Kamińska and Lewicki [6] obtained (using different techniques) a generalization of Theorem 1.1 in another direction -they characterized 1-complemented subspaces of finite codimension in sufficiently smooth Musielak-Orlicz sequence spaces, whose Orlicz function is sufficiently different from the square function.
We follow standard definitions and notations as may be found in [7,9]

Preliminary definitions
Orlicz spaces are one of the most natural generalizations of classical spaces L p . They were first considered by Orlicz in 1930s. Since then they were extensively studied by many authors, see, for example the monographs [7,18,3]. Below we recall the basic definitions and facts about Orlicz spaces that will be important for the present paper.  Note that M ∈ ∆ 2 does not imply that M * ∈ ∆ 2 . The Orlicz function M generates the modular defined for scalar sequences x = (x j ) j∈N by: The Orlicz sequence space ℓ M is the space of sequences x such that there exists λ > 0 with ρ M (λx) < ∞. If M ∈ ∆ 2 then ℓ M = {x : ρ M (λx) < ∞ for all λ ∈ R}. The Orlicz sequence space ℓ M is usually equipped with one of the two following equivalent norms: (1) the Luxemburg norm defined by: the Orlicz norm defined by: If M ∈ ∆ 2 then these norms are dual to each other in the following sense: We say that two Orlicz functions M 1 and M 2 are equivalent if there exist u 0 > 0, k, l > 0 such that for all u with |u| ≤ u 0 This condition is of importance since Orlicz spaces ℓ M 1 , ℓ M 2 are isomorphic if and only if the Orlicz functions M 1 , M 2 are equivalent. We note that if an Orlicz function M satisfies the condition ∆ 2 near zero then every Orlicz function M 1 equivalent to M also satisfies the condition ∆ 2 near zero.
Krasnoselskii and Rutickii proved the following characterization of the ∆ 2 -condition in terms of the right derivative M ′ of M.
In [15] we introduced another condition which on one hand is very similar to (2.3), but on the other hand is in its nature of "smoothness type", as we explain below.
Definition 2.4. Assume that the Orlicz function M is twice differentiable and that M satisfies the ∆ 2 −condition near zero. We say that M satisfies condition ∆ 2+ near zero if there exist constants β > 0 and u 0 ≥ 0 such that for all u ≤ u 0 Condition ∆ 2+ is of "smoothness type" in the following sense: (i) for every function M which satisfies condition ∆ 2 there exists an equivalent Orlicz function M 1 which does satisfy ∆ 2+ ; However, we do not know whether for every ε > 0 it is possible to choose M 1 so that it is (1 + ε)−equivalent with M, (ii) for every Orlicz function M which satisfies ∆ 2+ there exists an equivalent (even up to an arbitrary ε > 0) Orlicz function M 1 which does not satisfy ∆ 2+ .
We say that a Banach space X is smooth if every element x ∈ X has a unique norming functional x * ∈ X * , i.e. the functional with the property that x * 2 If M ∈ ∆ 2 then an Orlicz space ℓ M is smooth whenever M is differentiable everywhere. It is well known (see e.g. [3]) that any Orlicz function M can be "smoothed out", that is for any M there exists an equivalent Orlicz function M 1 such that M 1 is twice differentiable everywhere, M ′′ 1 is continuous on R and M ′′ 1 (u) > 0 for all u > 0. We recall here that a class of functions whose second derivative exists and is continuous on R is denoted by C 2 . Thus M 1 above belongs to C 2 . Moreover, given any ε > 0 it is possible to choose M 1 so that ℓ M and ℓ M 1 are (1 + ε)−isomorphic to each other [3].
Maleev and Troyanski [10] considered a stronger notion of smoothness in Orlicz spaces which guarantees the differentiability of the norm. We recall the relevant definitions and results.
Definition 2.5. [11] (cf. [9, p. 143]) To every Orlicz function M we associate the following Matuszewska-Orlicz index: Definition 2.6. [10] We say that an Orlicz function M belongs to the class AC k at zero if For an open set V ⊂ X, ϕ ∈ F k (V, Y ) means ϕ is k−times differentiable at every point of V . If (2.5) is fulfilled uniformly on f over a set W ⊂ V we shall say that ϕ is k−times uniformly differentiable over W and shall write ϕ ∈ UF k (W, Y ). We say that X is UF ksmooth if the norm in X belongs to UF k (S(X), R). We do not know whether in Theorem 2.9 it is possible for any ε > 0 to select M so that M and M are (1 + ε)−equivalent.
Next we recall that a Banach lattice X is called strictly monotone if x + y > x for all x, y ≥ 0, y = 0, in X.
An Orlicz space ℓ M with either the Luxemburg or the Orlicz norm is strictly monotone whenever M is strictly increasing on [0, ∞).

Tools
In this section we gather our main tools -facts about contractive projections and about disjointness in Orlicz spaces.
We will say that a projection P on a purely atomic Banach lattice X is an averaging projection if there exist mutually disjoint elements {u j } j∈J in X and functionals {u * j } j∈J in X * so that u * j (u k ) = 0 if j = k, u * j (u j ) = 1 for all j ∈ J and for each f ∈ X First we recall two abstract conditions that we introduced in [17] in our study of averaging projections in purely atomic Banach lattices.
Definition 3.1. [17] Let X be a Banach lattice and P : X → X be a linear operator on X. We say that the operator P is (1) semi band preserving if and only if for all f, g ∈ X, supp(P f ) ∩ supp(g) = ∅ implies that supp(P f ) ∩ supp(P g) = ∅. supp g ⊂ supp P f implies that supp P g ⊂ supp P f. In the above statement all set relations are considered modulo sets of measure zero. It is clear that all averaging projections are both semi band preserving and semi containment preserving. In [17] we proved that in fact in "nice" purely atomic Banach spaces either of semi band or semi containment preservation characterizes averaging projections among contractive projections. More precisely, we have: [17] Let X be a purely atomic strictly monotone Banach lattice and let P : X → X be a norm one projection which is semi band preserving or semi containment preserving. Then P is an averaging projection.
This theorem will be very useful for our considerations since in [15] we obtained conditions which partially describe disjointness and containment of supports of elements in Orlicz spaces. These conditions will enable us to verify that contractive projections in Orlicz sequence spaces are semi band preserving or semi containment preserving.
We note here that all theorems in [15] were formulated and proved for Orlicz function spaces L M , where M is an Orlicz function satisfying conditions ∆ 2 and ∆ 2+ near infinity. However to adapt to the case of Orlicz sequence spaces ℓ M , where M is an Orlicz function satisfying conditions ∆ 2 and ∆ 2+ near zero, the proofs require only very minor changes, if any. Thus in the following when we refer to the statements from [15] we will formulate them using ℓ M instead of L M , which is more appropriate for the present paper.
We stress that theorems in [15] are proven for Orlicz spaces equipped with the Luxemburg norm, and the analogs of most of the results from [15] are false in Orlicz spaces equipped with the Orlicz norm.   . We leave the details, which are easy but require cumbersome notation, to the interested reader.
Finally we recall a result from [13] which describes the form of two dimensional 1-complemented subspaces of Orlicz sequence spaces, when the two spanning elements have disjoint supports. (We say that a subspace is 1-complemented if it is the range of a projection P with P = 1.) This result will allow us to give a very detailed description of 1-complemented subspaces of any dimension of Orlicz sequence spaces.
Theorem 3.6. [13, Theorem 6.1] Let M be an Orlicz function satisfying condition ∆ 2 and ℓ M be a (real or complex) Orlicz sequence space equipped with either the Luxemburg or the Orlicz norm and let x, y ∈ ℓ M , be disjoint norm one elements such that span{x, y} is 1complemented in ℓ M . Then one of three possibilities holds: (1) card(supp x) < ∞ and |x i | = |x j | for all i, j ∈ supp x; or (2) there exists p, 1 ≤ p ≤ ∞, such that M(t) = Ct p for all t ≤ x ∞ ; or (3) there exists p, 1 ≤ p ≤ ∞, and constants C 1 , C 2 , γ ≥ 0 such that C 2 t p ≤ M(t) ≤ C 1 t p for all t ≤ x ∞ and such that, for all j ∈ supp x, In particular, it follows from Theorems 3.6 that in "most" Orlicz spaces the only 1complemented disjointly supported subspaces of any dimension are those spanned by a block basis with constant coefficients of some permutation of the original basis.
We are now ready for our main results. Proof. Since bounded functions with finite supports are linearly dense in ℓ M , to show that P is semi band preserving or semi containment preserving, respectively, it is enough to verify that (3.1) or (3.2), resp., are satisfied with the additional assumption that g is a bounded function and µ(supp g) < ∞.
As a consequence we obtain the characterization of contractive projections in Orlicz sequence spaces. Let ℓ M be the real Orlicz sequence space equipped with the Luxemburg norm and let P be a contractive projection on ℓ M . Then P is an averaging projection, i.e. there exist mutually disjoint elements {u j } j∈J in ℓ M and functionals {u * j } j∈J in (ℓ M ) * so that u * j (u k ) = 0 if j = k, u * j (u j ) = 1 for all j ∈ J and for each f ∈ ℓ M .
Moreover, one of the three possibilities holds: (1) card(supp u j ) < ∞ for each j ∈ J, and |(u j ) k | = |(u j ) l | for each k, l ∈ supp(u j ), j ∈ J.
Proof. Note first that either condition (i) or (ii) implies that ℓ M is smooth and that M ′ is a strictly increasing function on (0, ∞). Thus M is also strictly increasing on (0, ∞) and ℓ M is strictly monotone. Hence the fact that P is an averaging projection follows immediately from Corollary 3.2 and Proposition 4.2. The moreover part follows directly from [13, Theorem 6.1] (see Theorem 3.6). Indeed, since the elements {u j } j∈J are mutually disjoint, for any j 1 , j 2 ∈ J and any f ∈ ℓ M we have Thus the projection Q : ℓ M −→ span{u j 1 , u j 2 } defined by Qf = u * j 1 (f )u j 1 + u * j 2 (f )u j 2 , has Q = 1. Thus, by Theorem 3.6, conditions (1)-(3) in the statement of Theorem 4.3 are satisfied.
By duality we also obtain the description of contractive projections in real Orlicz sequence spaces equipped with the Orlicz norm. Let ℓ M be the real Orlicz sequence space equipped with the Orlicz norm and let P be a contractive projection on ℓ M . Then P has the form described in Theorem 4.3.
Proof. This follows from Theorem 4.3 by duality. Indeed, since M ∈ ∆ 2 , by (2.2) we have (ℓ M , · O M ) * = (ℓ M * , · M * ) and the dual projection P * is contractive in ℓ M * equipped with the Luxemburg norm. Further, either of the conditions (i * ) or (ii * ) implies that M * is smooth, so the only thing that needs to be verified is that condition (i * ) implies that M * satisfies condition (i) and condition (ii * ) implies that M * satisfies condition (ii) from Theorem 4.3.