Orlicz sequence space explained

In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below, under which it forms a Banach space. Orlicz sequence spaces generalize the

\ell _{p} spaces, and as such play an important role in functional analysis. Orlicz sequence spaces are particular examples of Orlicz spaces.

Definition

Fix

K \in\{ R , C \} so that K denotes either the real or complex scalar field. We say that a function M:[ 0 , inf ty)\to[ 0 , inf ty) is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with M( 0 ) =0 and $\backslash lim\_M(t)=\backslash infty$. In the special case where there exists b> 0 with M(t) =0 for all t\in[ 0 ,b] it is called degenerate.

In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. This implies

M(t)> 0 for all t> 0 .

For each scalar sequence

(a _{n)} ^{inf ty\in K} ^{n =1} ^{N} set\left\|(a _{n)} ^{inf ty\right\|} ^{M =inf \left\{\rho> 0 :\sum} ^{inf ty} ^{n =1} M(|a _{n|/\rho)\leqslant} 1 \right\}. We then define the Orlicz sequence space with respect to M , denoted \ell _{M} , as the linear space of all (a _{n)} ^{inf ty\in K} ^{n =1} ^{N} such that for some \rho> 0 , endowed with the norm \| ⋅ \| _{M} .

Two other definitions will be important in the ensuing discussion. An Orlicz function

M is said to satisfy the Δ₂ condition at zero whenever\limsup _{t\to} (M( 2 t))/(M(t)) < inf ty. We denote by h _{M} the subspace of scalar sequences (a _{n)} ^{inf ty\in\ell} ^{M} such that for all \rho> 0 .

Properties

The space

\ell _{M} is a Banach space, and it generalizes the classical \ell _{p} spaces in the following precise sense: when M(t) = t ^{p} , 1 \leqslant p< inf ty , then \| ⋅ \| _{M} coincides with the \ell _{p} -norm, and hence \ell _{M=\ell} _{p} ; if M is the degenerate Orlicz function then \| ⋅ \| _{M} coincides with the \ell _{infty} -norm, and hence \ell _{M=\ell} _{infty} in this special case, and h _{M=c} _{0} when M is degenerate.

In general, the unit vectors may not form a basis for

\ell _{M} , and hence the following result is of considerable importance.

Theorem 1. If

M is an Orlicz function then the following conditions are equivalent:

Two Orlicz functions

M and N satisfying the Δ₂ condition at zero are called equivalent whenever there exist are positive constants A,B,b> 0 such that AN(t)\leqslant M(t)\leqslant BN(t) for all t\in[ 0 ,b] . This is the case if and only if the unit vector bases of \ell _{M} and \ell _{N} are equivalent.\ell _{M} can be isomorphic to \ell _{N} without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)

Theorem 2. Let

M be an Orlicz function. Then \ell _{M} is reflexive if and only if\liminf _{t\to} (tM'(t))/(M(t)) > 1 and \limsup _{t\to} (tM'(t))/(M(t)) < inf ty .

Theorem 3 (K. J. Lindberg). Let

X be an infinite-dimensional closed subspace of a separable Orlicz sequence space \ell _{M} . Then X has a subspace Y isomorphic to some Orlicz sequence space \ell _{N} for some Orlicz function N satisfying the Δ₂ condition at zero. If furthermore X has an unconditional basis then Y may be chosen to be complemented in X , and if X has a symmetric basis then X itself is isomorphic to \ell _{N} .

Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space

\ell _{M} contains a subspace isomorphic to \ell _{p} for some 1 \leqslant p< inf ty .

Corollary. Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to

\ell _{p} for some 1 \leqslant p< inf ty .

Note that in the above Theorem 4, the copy of

\ell _{p} may not always be chosen to be complemented, as the following example shows.

Example (Lindenstrauss/Tzafriri). There exists a separable and reflexive Orlicz sequence space

\ell _{M} which fails to contain a complemented copy of \ell _{p} for any 1 \leqslant p\leqslant inf ty . This same space \ell _{M} contains at least two nonequivalent symmetric bases.

Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If

\ell _{M} is an Orlicz sequence space satisfying $\backslash liminf\_tM\text{'}(t)/M(t)=\backslash limsup\_tM\text{'}(t)/M(t)$ (i.e., the two-sided limit exists) then the following are all true.

Example. For each

1 \leqslant p< inf ty , the Orlicz function M(t) = t ^{p/(1-log} (t)) satisfies the conditions of Theorem 5 above, but is not equivalent to t ^{p} .

References

• • Lindenstrauss . Joram . Joram Lindenstrauss. Tzafriri . Lior. On Orlicz Sequence Spaces. Israel Journal of Mathematics. 10. 3. 379–390. September 1971. 10.1007/BF02771656 . free.
• Lindenstrauss . Joram . Joram Lindenstrauss. Tzafriri . Lior. On Orlicz Sequence Spaces. II. Israel Journal of Mathematics. 11. 4. 355–379. December 1972. 10.1007/BF02761463 . free.
• Lindenstrauss . Joram . Joram Lindenstrauss. Tzafriri . Lior. On Orlicz Sequence Spaces III. Israel Journal of Mathematics. 14. 4. 368–389. December 1973. 10.1007/BF02764715 . free.