Orlicz sequence space

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 $\mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}$ so that $\mathbb {K}$ denotes either the real or complex scalar field. We say that a function $M:[0,\infty )\to [0,\infty )$ is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with $M(0)=0$ and ${\textstyle \lim _{t\to \infty }M(t)=\infty }$. 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 $(a_{n})_{n=1}^{\infty }\in \mathbb {K} ^{\mathbb {N} }$ set

$\left\|(a_{n})_{n=1}^{\infty }\right\|_{M}=\inf \left\{\rho >0:\sum _{n=1}^{\infty }M(|a_{n}|/\rho )\leqslant 1\right\}.$

We then define the Orlicz sequence space with respect to $M$, denoted $\ell _{M}$, as the linear space of all $(a_{n})_{n=1}^{\infty }\in \mathbb {K} ^{\mathbb {N} }$ such that ${\textstyle \sum _{n=1}^{\infty }M(|a_{n}|/\rho )<\infty }$ for some $\rho >0$, endowed with the norm $\|\cdot \|_{M}$.

Two other definitions will be important in the ensuing discussion. An Orlicz function $M$ is said to satisfy the Δ₂ condition at zero whenever

$\limsup _{t\to 0}{\frac {M(2t)}{M(t)}}<\infty .$

We denote by $h_{M}$ the subspace of scalar sequences $(a_{n})_{n=1}^{\infty }\in \ell _{M}$ such that ${\textstyle \sum _{n=1}^{\infty }M(|a_{n}|/\rho )<\infty }$ 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 $M(t)=t^{p}$, $1\leqslant p<\infty$, then $\|\cdot \|_{M}$ coincides with the $\ell _{p}$-norm, and hence $\ell _{M}=\ell _{p}$; if $M$ is the degenerate Orlicz function then $\|\cdot \|_{M}$ 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:

1. $M$ satisfies the Δ₂ condition at zero, i.e. ${\textstyle \limsup _{t\to 0}M(2t)/M(t)<\infty }$.
2. For every $\lambda >0$ there exists positive constants $K=K(\lambda )$ and $b=b(\lambda )$ so that $M(\lambda t)\leqslant KM(t)$ for all $t\in [0,b]$.
3. ${\textstyle \limsup _{t\to 0}tM'(t)/M(t)<\infty }$ (where $M'$ is a nondecreasing function defined everywhere except perhaps on a countable set, where instead we can take the right-hand derivative which is defined everywhere).
4. $\ell _{M}=h_{M}$.
5. The unit vectors form a boundedly complete symmetric basis for $\ell _{M}$.
6. $\ell _{M}$ is separable.
7. $\ell _{M}$ fails to contain any subspace isomorphic to $\ell _{\infty }$.
8. $(a_{n})_{n=1}^{\infty }\in \ell _{M}$ if and only if ${\textstyle \sum _{n=1}^{\infty }M(|a_{n}|)<\infty }$.

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

$\liminf _{t\to 0}{\frac {tM'(t)}{M(t)}}>1\;\;$ and $\;\;\limsup _{t\to 0}{\frac {tM'(t)}{M(t)}}<\infty$.

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 $1\leqslant p<\infty$.

Corollary. Every infinite-dimensional closed subspace of a separable Orlicz sequence space contains a further subspace isomorphic to $\ell _{p}$ for some $1\leqslant p<\infty$.

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 $1\leqslant p\leqslant \infty$. 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 ${\textstyle \liminf _{t\to 0}tM'(t)/M(t)=\limsup _{t\to 0}tM'(t)/M(t)}$ (i.e., the two-sided limit exists) then the following are all true.

1. $\ell _{M}$ is separable.
2. $\ell _{M}$ contains a complemented copy of $\ell _{p}$ for some $1\leqslant p<\infty$.
3. $\ell _{M}$ has a unique symmetric basis (up to equivalence).

Example. For each $1\leqslant p<\infty$, 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; Tzafriri, Lior (1977), Classical Banach Spaces I, Sequence Spaces, ISBN 978-3-642-66559-2
• Lindenstrauss, Joram; Tzafriri, Lior (September 1971). "On Orlicz Sequence Spaces". Israel Journal of Mathematics. 10 (3): 379–390. doi:10.1007/BF02771656.
• Lindenstrauss, Joram; Tzafriri, Lior (December 1972). "On Orlicz Sequence Spaces. II". Israel Journal of Mathematics. 11 (4): 355–379. doi:10.1007/BF02761463.
• Lindenstrauss, Joram; Tzafriri, Lior (December 1973). "On Orlicz Sequence Spaces III". Israel Journal of Mathematics. 14 (4): 368–389. doi:10.1007/BF02764715.