Classification of Essentially Algebraic Composition Operators on Weighted Lorentz–Karamata Sequence Spaces

1 Introduction
Let be a -finite complete measure space and let be a measurable transformation, that is, for any . If for
each with , then is said to be non-singular.
Any non-singular measurable transformation induces a linear operator from into itself defined by

(1)

where denotes the linear space of all equivalence classes of -measurable functions on . Here we identify
any two functions that are equal -almost everywhere on .
Let be the class of all functions in that are finite -almost everywhere on . For , we define the distribution
function of on by

and the decreasing rearrangement of on by

The Lorentz space is the set of all classes of -measurable functions on such that the functional , where

Take , and . Then the Lorentz sequence space is the set of all sequences such that the functional ,
where

If we take , and is the weight function, then the corresponding Lorentz sequence space with weight is
denoted by and

Note that the Lorentz spaces are quasi-normed linear spaces and the functional is a norm if and only if
or .
For any -measurable set of finite measure, we have

For details about Lorentz spaces one can refer to [1, 5] and references therein.
Theorem 1.1 ([6, Theorem 2.3]). Let be a non-singular measurable transformation. Then induces a
bounded composition operator on , , if and only if there exists some constant such that for each .
Moreover,
For Orlicz spaces see [9, 13, 14] and for composition operators on Orlicz spaces one can refer to [8, 14]
and references therein. The work of this paper is motivated by the interesting work of Böttcher and
Heidler [3, 4] .
Definition 1.2 Let denote the ring of univariate polynomials with complex coefficients. A polynomial is
said to be monic if .
Definition 1.3 An operator on a Banach space is said to be algebraic if there is a non-zero polynomial
such that and will be called essentially algebraic if there is a non-zero polynomial such that is compact.
Definition 1.4 The monic polynomial of the least degree such that is zero is called the characteristic
polynomial of .
Definition 1.5 The monic polynomial of the least degree such that is compact is called the essentially
characteristic polynomial of .
Let and represent the characteristic and essentially characteristic polynomials associated with the linear
operator . Also for a polynomial , denotes the two sided ideal .
Let be the collection of all bounded operators on a Banach space and be the collection of all compact
operators on . is a Banach algebra under the operator norm and is a two sided ideal in . Let be the
natural map of onto the Calkin algebra . Let and . In case is finite dimensional, is isomorphic to and is
a closed subalgebra of . In case is finite dimensional, is isomorphic to and is a closed subalgebra of .
The essential characteristic polynomial of is actually the characteristic polynomial of the Calkin image of
in the Calkin algebra . See [3, 4] for more details about the above definitions and notations.

Algebraic and essentially algebraic composition operators on , are studied by Böttcher and Heidler [4]
and these results were extended to Orlicz spaces by Kumar and Kumar [5].
2 Weighted Lorentz–Karamata Sequence Spaces
Let , , and let be a weight sequence. Define the weighted counting measure

(2)

Let be a self-map. Since , every subset of measure zero is empty, so every self-map is nonsingular. The
composition operator is

Let be a slowly varying function. For

the weighted Lorentz–Karamata sequence space is the set of all sequences such that

(3)

where is the decreasing rearrangement of with respect to . When , is the usual weighted Lorentz
sequence space .
Assumption 2.1 The function is slowly varying at and at , locally bounded, and positive. We write .
Under Assumption 2.1, the space with the functional is a rearrangement-invariant Banach function
space with the Fatou property (see [11, 12, 10] for details). Throughout, denotes this Banach realization.
Assumption 2.2 The self-map is finite-to-one: for every , the fiber is finite.
Assumption 2.3 There is a constant such that for every periodic cycle of , and for all ,

Definition 2.4 For , , define the fundamental function analytically by

Lemma 2.5 For every measurable set of finite measure,
Proof. The distribution function of is for and for . Hence . Substitution in the norm gives the result. ◻
Proposition 2.6 Under Assumption 2.1, for , In particular, at both endpoints.
Proof. Write the integrand as . Since is slowly varying and , Karamata’s theorem (see [2, Theorem
1.5.11] ) gives

Taking -th roots yields the stated asymptotic. ◻
Corollary 2.7 Under Assumption 6, satisfies the doubling condition
Proof. By Proposition 2.6, the ratio tends to as and as . Since is continuous and positive on every
compact subinterval , the ratio is bounded above and below by positive constants there. Combining the
two endpoint limits with the compact-interval bound yields global constants with . ◻
Lemma 2.8 (Global scaling and inversion estimates). Under Assumption 2.1, the following hold.

1. (Scaling estimate.) For every there exists such that for all and all , Equivalently, with ,
2. (Inversion estimate.) For every there exists such that for all ,
   Proof. Since at both endpoints and is continuous and positive on , it satisfies Potter-type global bounds:
   for every there is a constant such that for all ,

and the reverse inequalities for . These bounds follow from the Uniform Convergence Theorem for
regularly varying functions together with the continuity and positivity of on compact subsets of (see [2,
Theorem1.5.6] and [1, Chapter 2] ).
(i) Fix and let . Applying the upper Potter bound with and (so ), we get

Since this holds for every , we obtain the stated estimate with .
(ii) Let be given. Choose and let be the constant from the global Potter bound. Suppose for some . If ,
then (since is increasing), so and is trivial for ; if , we may assume without loss of generality, since
otherwise the conclusion fails only for the trivial case which we exclude by choosing .
Assume therefore . Set . Applying the lower Potter bound with , , we get

Combining with ,

hence

Set

Then as , and . This proves (ii). ◻
Theorem 2.9 (Boundedness criterion). Let be a self-map. Then induces a bounded composition
operator if and only if where the supremum is taken over all with . Moreover, the operator norm is
equivalent to this supremum.
Proof. Since is a rearrangement-invariant Banach function space (see [11, 12] ), the composition
operator is bounded if and only if the measure is absolutely continuous with respect to with a bounded
Radon–Nikodym derivative in the associate norm. Concretely:
Suppose first that there is such that for every of finite measure. For and ,

Hence for all . Therefore

Substituting ,

By the doubling property of (equivalently, the slow variation of ), there is a constant depending only on
such that for all . Hence

so is bounded.
Conversely, suppose is bounded. For every of finite positive measure, and

while . Hence

which gives the stated supremum bound. ◻
3 Essential Orbit Structure
For a self-map , the set is called the orbit of . If the orbit of is a finite set, then we define the enter length
and the cycle length of the point as

and

If the orbit is infinite, then we take .
The maximal enter length of the self-map is denoted by and defined by

and the set of occurring periods or cycle lengths is

where

In case , we say that is an essential period of . The set of all essential periods is denoted by . The set of
all unessential periods is defined as

and the set of all periodic points with unessential periods is the set

The essential enter length for the self-map depends on the decreasing rearrangement and on . If the set

is finite, then we put . If is infinite, then we define as the minimal such that for every there is a finite set
with
(4)
Equivalently, the cofinite limit of the ratios vanishes:

in the sense that for every , all but finitely many satisfy the inequality. If there is no such that the above
condition is satisfied, then we put .
For details about the above definitions we refer to [3, 4] .
Lemma 3.1 The following are equivalent:
Proof. Suppose . Then is finite. Also

is a finite union of finite sets, hence finite.
Conversely, suppose and . If were infinite, then since for each and the sets are pairwise disjoint, the
union would be infinite. Contradiction. Hence is finite, so is finite. ◻
4 Norm Estimates for Characteristic Functions
Lemma 4.1 Let be disjoint with , , . Put , . Then for , In particular, if , then
Proof. The decreasing rearrangement of is

Substituting this into the Lorentz–Karamata norm gives the stated formula. ◻
Remark 4.2 Because is doubling (Corollary 2.7), Lemma 4.1 implies that normalized differences of
disjoint characteristic functions are bounded away from zero uniformly when and are comparable.
Explicitly, if , then

independently of .
Lemma 4.3 (Compression estimate). Let be a set of atoms, and let . Let be supported on , and suppose
that Then where is arbitrary and depends only on .
Proof. For , set . Then

Hence for all . Therefore

Substituting gives

By Potter’s bound, for and any . Hence

Taking -th roots gives the result. ◻
5 Disjointness of Preimage Trees
Lemma 5.1 Let be aperiodic, i.e. is not periodic. Then for ,

Proof. Suppose with . Then . Applying to the first equality gives . Hence , so is periodic with period ,
contradiction. ◻
Lemma 5.2 Suppose satisfies Assumption 2.2 and is infinite. Let . Then for every infinite set there
exists a sequence such that the sets are pairwise disjoint.
Proof. We construct the sequence inductively. Since is infinite, choose arbitrarily.
Suppose have been chosen. Define the finite set

Each is a single point (or empty), and each is finite by Assumption 2.2. Since there are finitely many
and finitely many , the set is finite. Since is infinite, choose .
We claim that the sets for and are pairwise disjoint. Suppose

for some and . Then and .
If , then . So with , hence , contradiction.
If , then . So with , hence , contradiction.
Therefore the sets are pairwise disjoint. ◻
6 Essential Characteristic Polynomial
Lemma 6.1 Let be periodic with cycle length , so that are distinct and . Let . For , put Let , . Then
Moreover, If Assumption 2.3 holds, then Finally, the algebraic identity holds. More precisely, if with ,
then .
Proof. We have

On the cycle, the coefficient at the point is precisely . Since the points are distinct, the functions are
disjointly supported. Hence

where is an index at which the maximum is attained. Taking norms and using monotonicity gives the first
bound.
For the second bound, by Assumption 2.3 and Corollary 2.7, iterating the doubling inequality a finite
number of times yields uniformly in , where depends only on and the doubling constant.
For the algebraic identity, divide by with remainder:

Write . Since for all , the coefficient of in equals the sum of the coefficients with , which is . Hence iff all
, i.e. iff . ◻
Lemma 6.2 Suppose . Then there exists a sequence of periodic points with period such that the cycles
are pairwise disjoint. Moreover, if is a polynomial such that , then the sequence satisfies and there is a
constant , independent of , such that for all . Hence cannot be compact.

Proof. Since , the set is infinite. We extract a sequence of pairwise disjoint -cycles greedily: choose ;
since is infinite, choose outside the finite cycle of ; continue inductively.
For the periodic point , the entire inverse orbit is contained in the cycle:

Hence the supports of are contained in the disjoint cycles.
For each , Lemma 6.1 and Assumption 2.3 give

where . If , then not all are zero, so , and the lower bound is uniform in with constant .
Since the cycles are pairwise disjoint, the functions and are supported on disjoint sets for . Applying
Lemma 4.1 and Remark 4.2 with (after normalization), there is a constant , independent of , such that

Thus the sequence has no convergent subsequence, so is not compact. ◻
Lemma 6.3 Suppose . Then there exist infinitely many such that If with , then along a suitable
sequence, Hence cannot be compact. Therefore, if is compact, then for every ; equivalently,
Proof. Since , there is an and an infinite set such that

Otherwise the cofinite limit defining would vanish for this , contradicting .
Apply Lemma 5.2 with equal to the degree of and with the infinite set , obtaining a sequence such that
the sets

are pairwise disjoint.
By Lemma 5.1, for each fixed the sets for are pairwise disjoint. Lemma 5.2 ensures disjointness across
different .
For each ,

The term with has coefficient and is supported on , which is disjoint from all other terms. Hence

This is a uniform lower bound. The sequence is bounded (norm 1), and the images have disjoint
supports, so by the lattice property,

Thus no subsequence converges, and is not compact. ◻
7 The Key Compactness Lemma
Lemma 7.1 Let be a self-map satisfying Assumptions 2.2 and 2.3.Assume that and that . If then is
compact.

Proof. Decompose with

Both sets are forward -invariant: if , then and hence , so ; similarly for . Since is a Banach lattice with
order-continuous norm, the multiplication projections

are bounded, and

Periodic part. Decompose by cycle length. For , let

Then

Since , this is a finite direct sum of invariant subspaces.
On with , we have , so . Hence

On with , the set is finite, so is finite-dimensional, and every operator on it is compact. Therefore is
compact.
Aperiodic part. We show that is compact. Since

and the product is bounded while compact operators form a two-sided ideal, it suffices to prove that is
compact.
For , define

By the definition of , the set is finite for every .
Let be the projection onto , and set on . The projection has finite rank.
We estimate for . Write with (by definition of , is supported on ). For , we have

By Lemma 2.8(ii), there exists with as such that

Set for sufficiently small. By Lemma 4.3,

Hence

Therefore

where the first term has finite rank and the second has norm tending to zero as . Hence is a norm limit of
finite-rank operators and is therefore compact.
Consequently, is compact on both and , hence on all of . ◻
Main Classification Theorem
Theorem 25. Let be a self-map satisfying Assumptions 2.2 and 2.3. Assume that and that . Then the
following are equivalent:
Moreover, if is finite-dimensional, then where with .
Proof. The equivalence of the second and third conditions is Lemma 3.1.
Necessity. Suppose . Then there is a nonzero polynomial such that is compact. By Lemma 6.2, must
be divisible by for every . By Lemma 6.3, must be divisible by for every . Hence

If either or , then no nonzero polynomial can satisfy these divisibility conditions, contradiction. Hence
and . By Lemma 3.1, this is equivalent to the second condition.
Sufficiency. Suppose and . By Lemma 7.1, the polynomial

satisfies . Hence , so is algebraic in the Calkin algebra. Consider the algebra homomorphism

Since , the kernel is a nonzero principal ideal of , say . By the necessity part, every annihilating
polynomial is divisible by

so this polynomial has the least degree among the annihilators and is thus the monic generator of the
kernel. Hence , and by the first isomorphism theorem,

 ◻
Corollary 8.2 Under the hypotheses of Theorem 8.1, if and only if Equivalently, is compact if and only
if the minimal annihilating polynomial of in the Calkin algebra is .
Proof. is compact iff iff the minimal annihilating polynomial is . By Theorem 8.1, iff , , and . ◻
9 Invariance Under Slowly Varying Perturbation
Proposition 9.1 Let . For positive sequences with and , The same holds if .
Proof. By Proposition 2.6,

Hence

By the Potter bounds for slowly varying functions [2, Theorem 1.5.6], for every there is a constant such
that

Choosing yields

whenever . The converse follows from the reverse Potter bound

for some . ◻
Proposition 9.2. Let . For positive sequences with for all , there are constants such that for all .
Proof. By the doubling condition (Corollary 2.7), iterating the inequality a finite number of times gives for
any fixed . Similarly, iterating the lower bound gives . Applying these with and to and yields the
result. ◻
Theorem 9.3 Let be a self-map satisfying Assumptions 2.2 and 2.3, and assume that for every and
every sequence , Then where is the essential enter length in the classical Lorentz space .
Consequently, the classification of essentially algebraic composition operators on is identical to that on .
Proof. The hypothesis states precisely that the cofinite limit defining vanishes if and only if the cofinite
limit defining vanishes, for every . Hence the minimal such is the same in both cases. The remaining
combinatorial data , , and are independent of the space. Therefore the classification theorem and the
compactness corollary are the same. ◻
Corollary 9.4 The hypothesis of Theorem 9.3 is satisfied if, for every , the ratios either tend to , or satisfy
, as along .
Proof. This follows from Propositions 9.1 and 9.2. ◻
Remark 9.5 The slowly varying function does not change the zero/nonzero nature of the asymptotic
ratios that determine compactness and essential algebraicity. The norm itself changes when changes,
but the algebraic classification of composition operators remains the same as in the classical Lorentz
case, under the stated regular-variation hypotheses.
10 Examples
Example 10.1 (Pure periodic case). Let consist of infinitely many cycles of length . Then

Hence , and

Thus

Example 10.2 (Infinite essential enter length). Let

Then is finite-to-one: , and for . With counting measure,

so

Hence the limit defining does not vanish for any , so

This example shows that finite-to-one maps can have infinite essential enter length.
Example 10.3 (Finite essential enter length). Let for , and for not a power of . Then , , , and so on. The
point is fixed, , , , etc.
Define the weight

Then is finite-to-one. For ,

so

For , the ratio is . Hence the cofinite limit defining vanishes precisely for , so

To see that is bounded, we check the criterion of Theorem 2.9. For a singleton with not a power of ,
and , so the ratio is . For ,

For , so

For a general set , write where contains non-powers of and contains powers of . Then

Since for all and , we obtain

Hence the ratio for all , and by monotonicity of , the supremum in Theorem 2.9 is finite. Therefore is
bounded on .
The essential characteristic polynomial is

since (all fixed points have cycle length , and there are infinitely many of them).
Example 10.4 (Slowly varying perturbation). Let

Then , and

For the explicit sequences

we have and

So the slowly varying factor does not change the zero/nonzero nature of the limit.

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