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 ${\displaystyle \ell _{p}}$ spaces, and as such play an important role in functional analysis.

Definition

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

In what follows, unless otherwise stated we'll assume all Orlicz functions are nondegenerate. This implies ${\displaystyle M(t)>0}$ for all ${\displaystyle t>0}$.

For each scalar sequence ${\displaystyle (a_{n})_{n=1}^{\infty }\in \mathbb {K} ^{\mathbb {N} }}$ set

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

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

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

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

We denote by ${\displaystyle h_{M}}$ the subspace of scalar sequences ${\displaystyle (a_{n})_{n=1}^{\infty }\in \ell _{M}}$ such that ${\displaystyle \textstyle \sum _{n=1}^{\infty }M(|a_{n}|/\rho )<\infty }$ for all ${\displaystyle \rho >0}$.

Properties

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

In general, the unit vectors may not form a basis for ${\displaystyle \ell _{M}}$, and hence the following result is of considerable importance.

Theorem 1. If ${\displaystyle M}$ is an Orlicz function then the following conditions are equivalent:

(i) ${\displaystyle M}$ satisfies the Δ₂ condition at zero, i.e. ${\displaystyle \textstyle \limsup _{t\to 0}M(2t)/M(t)<\infty }$.

(ii) For every ${\displaystyle \lambda >0}$ there exists positive constants ${\displaystyle K=K(\lambda )}$ and ${\displaystyle b=b(\lambda )}$ so that ${\displaystyle M(\lambda t)\leqslant KM(t)}$ for all ${\displaystyle t\in [0,b]}$.

(iii) ${\displaystyle \textstyle \limsup _{t\to 0}tM'(t)/M(t)<\infty }$ (where ${\displaystyle M'}$ 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).

(iv) ${\displaystyle \ell _{M}=h_{M}}$.

(v) The unit vectors form a boundedly complete symmetric basis for ${\displaystyle \ell _{M}}$.

(vi) ${\displaystyle \ell _{M}}$ is separable.

(vii) ${\displaystyle \ell _{M}}$ fails to contain any subspace isomorphic to ${\displaystyle \ell _{\infty }}$.

(viii) ${\displaystyle (a_{n})_{n=1}^{\infty }\in \ell _{M}}$ if and only if ${\displaystyle \textstyle \sum _{n=1}^{\infty }M(|a_{n}|)<\infty }$.

Two Orlicz functions ${\displaystyle M}$ and ${\displaystyle N}$ satisfying the Δ₂ condition at zero are called equivalent whenever there exist are positive constants ${\displaystyle A,B,b>0}$ such that ${\displaystyle AN(t)\leqslant M(t)\leqslant BN(t)}$ for all ${\displaystyle t\in [0,b]}$. This is the case if and only if the unit vector bases of ${\displaystyle \ell _{M}}$ and ${\displaystyle \ell _{N}}$ are equivalent.

${\displaystyle \ell _{M}}$ can be isomorphic to ${\displaystyle \ell _{N}}$ without their unit vector bases being equivalent. (See the example below of an Orlicz sequence space with two nonequivalent symmetric bases.)

Theorem 2. Let ${\displaystyle M}$ be an Orlicz function. Then ${\displaystyle \ell _{M}}$ is reflexive if and only if

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

Theorem 3 (K. J. Lindberg). Let ${\displaystyle X}$ be an infinite-dimensional closed subspace of a separable Orlicz sequence space ${\displaystyle \ell _{M}}$. Then ${\displaystyle X}$ has a subspace ${\displaystyle Y}$ isomorphic to some Orlicz sequence space ${\displaystyle \ell _{N}}$ for some Orlicz function ${\displaystyle N}$ satisfying the Δ₂ condition at zero. If furthermore ${\displaystyle X}$ has an unconditional basis then ${\displaystyle Y}$ may be chosen to be complemented in ${\displaystyle X}$, and if ${\displaystyle X}$ has a symmetric basis then ${\displaystyle X}$ itself is isomorphic to ${\displaystyle \ell _{N}}$.

Theorem 4 (Lindenstrauss/Tzafriri). Every separable Orlicz sequence space ${\displaystyle \ell _{M}}$ contains a subspace isomorphic to ${\displaystyle \ell _{p}}$ for some ${\displaystyle 1\leqslant p<\infty }$.

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

Note that in the above Theorem 4, the copy of ${\displaystyle \ell _{p}}$ 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 ${\displaystyle \ell _{M}}$ which fails to contain a complemented copy of ${\displaystyle \ell _{p}}$ for any ${\displaystyle 1\leqslant p\leqslant \infty }$. This same space ${\displaystyle \ell _{M}}$ contains at least two nonequivalent symmetric bases.

Theorem 5 (K. J. Lindberg & Lindenstrauss/Tzafriri). If ${\displaystyle \ell _{M}}$ is an Orlicz sequence space satisfying ${\displaystyle \textstyle \liminf _{t\to 0}tM'(t)/M(t)=\limsup _{t\to 0}tM'(t)/M(t)}$ (i.e., the two-sided limit exists) then the following are all true.

(i) ${\displaystyle \ell _{M}}$ is separable.

(ii) ${\displaystyle \ell _{M}}$ contains a complemented copy of ${\displaystyle \ell _{p}}$ for some ${\displaystyle 1\leqslant p<\infty }$.

(iii) ${\displaystyle \ell _{M}}$ has a unique symmetric basis (up to equivalence).

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

References

• Lindenstrauss, J., and L. Tzafriri. Classical Banach Spaces I, Sequence Spaces (1977), ISBN 978-3-642-66559-2.
• Lindenstrauss, J., and L. Tzafriri. "On Orlicz Sequence Spaces," Israel Journal of Mathematics 10:3 (Sep 1971), pp379-390.
• Lindenstrauss, J., and L. Tzafriri. "On Orlicz sequence spaces. II," Israel Journal of Mathematics 11:4 (Dec 1972), pp355-379.
• Lindenstrauss, J., and L. Tzafriri. "On Orlicz sequence spaces III," Israel Journal of Mathematics 14:4 (Dec 1973), pp368-389.