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 $M:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is an Orlicz function if it is continuous, nondecreasing, and (perhaps nonstrictly) convex, with $M(0)=0$ and $\underset{t\to \mathrm{\infty }}{\lim }M(t)=\mathrm{\infty }$. 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 $(a_n{)}_{n=1}^{\mathrm{\infty }}\in {\mathbb{𝕂}}^{\mathbb{\mathbb{N}}}$ set

${\Vert (a_n{)}_{n=1}^{\mathrm{\infty }}\Vert }_M=\inf \{\rho >0:\sum _{n=1}^{\mathrm{\infty }}M(|a_n|/\rho )⩽1\}.$

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

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

$\underset{t\to 0}{\mathrm{lim\; sup}}\frac{M(2t)}{M(t)}<\mathrm{\infty }.$

We denote by $h_M$ the subspace of scalar sequences $(a_n{)}_{n=1}^{\mathrm{\infty }}\in {\ell }_M$ such that ${\sum }_{n=1}^{\mathrm{\infty }}M(|a_n|/\rho )<\mathrm{\infty }$ 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 $M(t)=t^p$, $1⩽p<\mathrm{\infty }$, then $\Vert \cdot {\Vert }_M$ coincides with the ${\ell }_p$-norm, and hence ${\ell }_M={\ell }_p$; if $M$ is the degenerate Orlicz function then $\Vert \cdot {\Vert }_M$ coincides with the ${\ell }_{\mathrm{\infty }}$-norm, and hence ${\ell }_M={\ell }_{\mathrm{\infty }}$ 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. $\underset{t\to 0}{\mathrm{lim\; sup}}M(2t)/M(t)<\mathrm{\infty }$.
2. For every $\lambda >0$ there exists positive constants $K=K(\lambda )$ and $b=b(\lambda )$ so that $M(\lambda t)⩽KM(t)$ for all $t\in [0,b]$.
3. $\underset{t\to 0}{\mathrm{lim\; sup}}t{M}^{\prime }(t)/M(t)<\mathrm{\infty }$ (where $M^{\prime }$ 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 ${\ell }_{\mathrm{\infty }}$.
8. $(a_n{)}_{n=1}^{\mathrm{\infty }}\in {\ell }_M$ if and only if ${\sum }_{n=1}^{\mathrm{\infty }}M(|a_n|)<\mathrm{\infty }$.

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 $AN(t)⩽M(t)⩽BN(t)$ 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

$\underset{t\to 0}{\mathrm{lim\; inf}}\frac{t{M}^{\prime }(t)}{M(t)}>1$ and $\underset{t\to 0}{\mathrm{lim\; sup}}\frac{t{M}^{\prime }(t)}{M(t)}<\mathrm{\infty }$.

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 $1⩽p<\mathrm{\infty }$.

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

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 $1⩽p⩽\mathrm{\infty }$. 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 $\underset{t\to 0}{\mathrm{lim\; inf}}t{M}^{\prime }(t)/M(t)=\underset{t\to 0}{\mathrm{lim\; sup}}t{M}^{\prime }(t)/M(t)$ (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 $1⩽p<\mathrm{\infty }$.
3. ${\ell }_M$ has a unique symmetric basis (up to equivalence).

Example. For each $1⩽p<\mathrm{\infty }$, 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

• Classical Banach Spaces I, Sequence Spaces, 1977, ISBN 978-3-642-66559-2 
• "On Orlicz Sequence Spaces". Israel Journal of Mathematics 10 (3): 379–390. September 1971. doi:10.1007/BF02771656. 
• "On Orlicz Sequence Spaces. II". Israel Journal of Mathematics 11 (4): 355–379. December 1972. doi:10.1007/BF02761463. 
• "On Orlicz Sequence Spaces III". Israel Journal of Mathematics 14 (4): 368–389. December 1973. doi:10.1007/BF02764715.