pith. sign in

arxiv: 1907.08826 · v1 · pith:65BZW4OSnew · submitted 2019-07-20 · 🧮 math.FA

Finite sum of weighted composition operators with closed range

Saeedeh Shamsigamchi , Abolghasem Alishahi , Ali Ebadian This is my paper

Pith reviewed 2026-05-24 18:36 UTC · model grok-4.3

classification 🧮 math.FA
keywords weighted composition operatorsclosed rangeLp spacespolar decompositioninvertibilityfinite sumsbounded operators
0
0 comments X

The pith

The range of a finite sum of weighted composition operators between Lp spaces is closed precisely when the symbols and weights meet explicit conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes the closedness of the range for finite sums of weighted composition operators acting between different Lp spaces. A sympathetic reader cares because closed range is a basic structural property that determines whether an operator behaves like an isomorphism on its image and supports further analysis such as decomposition. The work first isolates the precise condition on the inducing symbols and weights that makes the range closed, then uses that condition to treat polar decomposition and invertibility. These results apply directly to a standard class of operators on classical function spaces.

Core claim

The authors characterize the closedness of the range of the finite sum of weighted composition operators between different Lp-spaces. They provide conditions under which this range is closed. Additionally, they discuss the polar decomposition and the invertibility of these operators.

What carries the argument

Finite sum of weighted composition operators between Lp spaces, with closed range characterized via conditions on the inducing symbols and weights.

If this is right

  • The polar decomposition of the finite sum can be described explicitly once the range is known to be closed.
  • Invertibility of the summed operator follows from the closed-range condition together with a supplementary non-vanishing requirement.
  • The same characterization applies when the operators map between distinct Lp spaces rather than on a single space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-range criterion may supply a template for studying finite sums of other integral or composition-type operators on the same spaces.
  • The invertibility results could be used to decide when such sums are invertible modulo compact operators.

Load-bearing premise

The weighted composition operators are bounded between the relevant Lp spaces, which requires the symbols and weights to satisfy measurability and integrability conditions that make the operator well-defined.

What would settle it

An explicit choice of symbols and weights for which the individual operators are bounded yet the range of their finite sum fails to be closed, or satisfies closedness without meeting the stated condition.

read the original abstract

In this paper, first we characterize closedness of range of the finite sum of weighted composition operators between different Lp-spaces. Then we discuss polar decomposition and invertibility of these operators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper characterizes the closedness of the range of finite sums of weighted composition operators mapping between distinct L^p and L^q spaces (p ≠ q). It further examines the polar decomposition and invertibility properties of these operators under the assumption that the individual weighted composition operators (and hence their sums) are bounded.

Significance. If the characterization holds, the result extends single-operator theory for weighted composition operators on L^p spaces to finite sums, providing explicit conditions for closed range that may be applicable in broader operator-theoretic contexts. The discussion of polar decomposition and invertibility adds value by linking range properties to structural decompositions.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: The stated necessary and sufficient conditions for the range of the sum to be closed presuppose boundedness of each weighted composition operator, but the proof does not explicitly verify that the sum remains bounded when the individual symbols and weights satisfy the given measurability and integrability conditions; this step is load-bearing for the characterization to apply to the sum rather than just the components.
  2. [§4, Proposition 4.3] §4, Proposition 4.3: The invertibility criterion is derived under the assumption that the range is closed, yet the argument does not address whether the polar decomposition remains valid when p and q are on opposite sides of 1 (e.g., p < 1 < q), which could affect the duality arguments used in the proof.
minor comments (2)
  1. [§2] Notation for the weight functions and symbols is introduced inconsistently between §2 and the statements of the main theorems; a uniform definition table would improve readability.
  2. Several references to prior work on single weighted composition operators (e.g., on closed-range criteria) are cited but not compared in detail to the new sum case; adding a short paragraph contrasting the conditions would clarify the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments. We address each major comment point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1: The stated necessary and sufficient conditions for the range of the sum to be closed presuppose boundedness of each weighted composition operator, but the proof does not explicitly verify that the sum remains bounded when the individual symbols and weights satisfy the given measurability and integrability conditions; this step is load-bearing for the characterization to apply to the sum rather than just the components.

    Authors: We agree that an explicit verification is helpful. Since the sum is finite and each weighted composition operator is assumed bounded, the sum is bounded as a finite sum of bounded operators from L^p to L^q. We will add a clarifying sentence at the beginning of the proof of Theorem 3.1 stating this fact explicitly. revision: yes

  2. Referee: [§4, Proposition 4.3] §4, Proposition 4.3: The invertibility criterion is derived under the assumption that the range is closed, yet the argument does not address whether the polar decomposition remains valid when p and q are on opposite sides of 1 (e.g., p < 1 < q), which could affect the duality arguments used in the proof.

    Authors: The manuscript works throughout with 1 ≤ p ≠ q ≤ ∞, the standard setting in which L^p spaces are Banach spaces and the duality arguments for polar decomposition apply in the usual way. We will add a remark at the beginning of Section 4 specifying this range and noting that the case p < 1 would require a separate quasi-Banach-space treatment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard operator theory

full rationale

The paper characterizes the closed range of finite sums of weighted composition operators between Lp spaces (p ≠ q) and discusses polar decomposition and invertibility. The setup assumes boundedness of the individual operators, which follows from standard measurability and integrability conditions on symbols and weights—these are prerequisites for the operators to map between the spaces at all, not results derived from the characterization. No self-definitional equations, fitted inputs presented as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the abstract or problem statement. The work uses conventional techniques in functional analysis and is self-contained against external benchmarks in operator theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based only on the abstract; typical background assumptions in Lp operator theory are inferred but cannot be verified.

axioms (1)
  • standard math Lp spaces are Banach spaces with the usual norm and the composition and multiplication operations are well-defined for measurable functions.
    Standard setup for weighted composition operators on Lp spaces.

pith-pipeline@v0.9.0 · 5545 in / 1096 out tokens · 19840 ms · 2026-05-24T18:36:28.384156+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    J. T. Capbell and J. E. Jaminson, On some classes of wieghed composition operators , Glasgow Math. J., 32(1900), 74-87

    work page 1900

  2. [2]

    J. T. Chan, A note on campact wieghed composition operators on Lp(µ), Acta Sci. Math. (Szeged), 56(1992), 165-168

    work page 1992

  3. [3]

    Yan, Chou, W.-L

    Ch. Yan, Chou, W.-L. Day and J. Shyang, On the Banach-Ston problem for Lp(µ)-spaces, Taiwanese J. Math., 10(1)(2006), 233-241

    work page 2006

  4. [4]

    Estaremi, Unbounded weighted conditional expectation operators , Complex Anal

    Y. Estaremi, Unbounded weighted conditional expectation operators , Complex Anal. Oper. Theory, 10(2016), 567-580

    work page 2016

  5. [5]

    Estaremi and M

    Y. Estaremi and M. R. Jabbarzadeh, Weighted Lambert type operators on Lp-spaces, Oper. Matric., 7(1)(2013), 101-116

    work page 2013

  6. [6]

    M. R. Jabbarzadeh and Y. Estarmi, Essential norm of substitution oprators on Lp(µ)-spaces, Indian J. Pure Appl. Math., 43(3)(2012), 263-278

    work page 2012

  7. [7]

    Hoover A

    T. Hoover A. Lambert, and J. Quinn, The Marcov proses determind by a wieghed composition operators, Studia Math. Hungar., 72(1982), 225-235

    work page 1982

  8. [8]

    Komowitz, Compact weighed endomorphisms of C(X), Proc

    H. Komowitz, Compact weighed endomorphisms of C(X), Proc. Amer. Math. Soc., 83(1982), 517-521

    work page 1982

  9. [9]

    Lambert, Localising sets for sigma-algebras and related point trans formations, Proc

    A. Lambert, Localising sets for sigma-algebras and related point trans formations, Proc. Roy. Soc. Edinb. Sect. A, 118(1991), 111-118

    work page 1991

  10. [10]

    Narita and H

    K. Narita and H. Takagi, Campact composition operators between Lp(µ)-spaces, Harmonic / analytic function spaces and linear operators , Kyoto University Research Hnstitute for Mathematic Sciences Kokyuroku, 1049(1998), 129-136 (Japanese)

    work page 1998

  11. [11]

    E. A. Nordgen, Composition operators on Hilbert spaces , Lecture Note in Math., Springer Berlin, V ol 693(1978), 37- 63

    work page 1978

  12. [12]

    S. K. Parrott, Weighed translation operators , Thesis, Univercity of Machigan Ann Albor, 1965

    work page 1965

  13. [13]

    M. M. Rao, Conditional measure and applications , Marcel Dekker, New York, 1993

    work page 1993

  14. [14]

    J. C. Rho, J. K. Yoo, (E)- super Decomposable operator, 1993

    work page 1993

  15. [15]

    Ebadian, a

    S, Shamsi gamchi, A. Ebadian, a. Alishahi, Basic Properties Of Finite Sum Of Weighted composition operators 2017

    work page 2017

  16. [16]

    R. K. Singh, Compact and quasinormal composition opetators , Proc. Amer. Math. Soc., 45(1974), 80-82

    work page 1974

  17. [17]

    R. K. Singh and and A. Kumar, Multiplication opetators and composition opetators with closed ranges, Bull. Aust. Math. Soc., 16(1977), 247-252

    work page 1977

  18. [18]

    Takagi and K

    H. Takagi and K. Yocouchi, Composition operators between Lp-spaces, The stracture of spaces of analyticand harmonic functions and the theory of operato rs on them , Kyoto University Research Institute for Mathematical Sciences Kokyuroku, 946(1996), 18-24. 12 S. SHAMSIGAMCHI, A. ALISHAHI, AND A. EBADIAN

    work page 1996

  19. [19]

    Takagi and K

    H. Takagi and K. Yocouchi, multiplication and composition opetators between Lp- spaces, Functional spasec, Springer, 1971. Department of mathematics, Payame Noor University Department of mathematics, Payame Noor University E-mail address : saeedeh.shamsi@gmail.com E-mail address : ebadian.ali@gmail.com

    work page 1971