Operator Theory

This paper explores a version of the classical Cesàro integral operator for the Lebesgue space L 2 (0, 1) where we discuss its norm, adjoint, spectral properties, and invariant subspaces. An important tool will be semigroups of weighted composition operators on L 2 (0, 1).

This paper explores a version of the classical Cesàro integral operator for the Lebesgue space L p (0, 1) where we discuss its norm, spectral properties, cyclicity, and invariant subspaces. The spectrum of the Cesàro operator will be a crescent domain whose geometry depends on p. An important tool will be semigroups of weighted composition operators on L p (0, 1).

This paper establishes a novel topological and combinatorial framework for the construction of the real continuum $\mathbb{R}$ and the spectral characterization of the Riemann Zeta zeros. By replacing classical Cauchy sequences with asymptotic ratios of purified integer partition states—defined here as Kaleidoscopic Cauchy Sequences—we derive the real continuum as an emergent thermodynamic phase generated from discrete geometry. We further extend this framework to the spectral realization of Riemann Zeta zeros, identifying them as spectral points of perfect destructive interference within the $A_{k-1}$ root lattice, unified by the Kaleidoscopic Algebra. Exact convergence and strict confinement of these zeros to the critical line are guaranteed by the annihilation of low-dimensional thermodynamic noise through Weyl group reflections. This unified architecture provides a rigorous algorithmic solution to both the Riemann Hypothesis and the Generalized Riemann Hypothesis (GRH). Furthermore, we formulate the Universal Caleidoscopic Filter Theorem and provide its complete proof, demonstrating its absolute universality by deriving the laws of acoustic diatonic asymmetry, wave interference, diffraction, Planck's Black Body Radiation, Einstein's Photoelectric Effect, Heisenberg's Uncertainty Principle, atomic quantization, quantum entanglement, fermionic Pauli exclusion, Bekenstein-Hawking entropy, and gravitational metric tensors directly from the discrete combinatorial geometry of the partition lattice.

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In this work, we introduce and study the S-pseudospectra of linear operators defined by nonstrict inequality in a Hilbert space. Inspired by A. B öttcher's result [3], we prove that the S-resolvent norm of bounded linear operators is not constant in any open set of the S-resolvent set. Beside, we find a characterization of the S-pseudospectrum of bounded linear operator by means the S-spectra of all perturbed operators with perturbations that have norms strictly less than ε.

Reasonable efforts have been made to publish reliable data and information, but the authors, editors, and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors, editors, and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint.

In this note, we mainly study the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities. By using the generalized Hölder inequality for symmetric gauge functions, we obtain a more general version of a norm inequality for positive real numbers α, β, γ and r satisfying 1 α + 1 β = 1 γ and r ⩾ max{ 1 α , 1 β }. The obtained result generalizes the inequalities of

In this note, we mainly study the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities. By using the generalized Hölder inequality for symmetric gauge functions, we obtain a more general version of a norm inequality for positive real numbers α, β, γ and r satisfying 1 α + 1 β = 1 γ and r ⩾ max{ 1 α , 1 β }. The obtained result generalizes the inequalities of

This paper establishes a formal operator-theoretic foundation for Zero Logic as a conservative extension of classical propositional logic. Unlike traditional propositional reasoning, which represents propositions through scalar truth assignments, Zero Logic models propositions as bounded epistemic state operators. We introduce a Boolean embedding framework that maps classical truth valuations into a spectral operator space while preserving their original semantics. A sequence of structural results is developed, including the Boolean Embedding Theorem, Spectral Completeness Corollary, Strict Generalization Theorem, and Spectral Conservativity Theorem. These results culminate in the Fundamental Extension Theorem of Zero Logic, proving that the Boolean valuation space is a proper subset of the Zero Logic operator space while classical semantics remain invariant under embedding. We then extend the framework with a rigorous proof calculus containing spectral safety invariants: Spectral Modus Ponens (S-MP), Spectral Modus Tollens (S-MT), and Epistemic Dampening for conjunction, together with a Soundness Theorem. Furthermore, we develop Probabilistic-Temporal Zero Logic (PT-ZL) through continuous-time epistemic decay dynamics with a proof of monotonic viability decay, Monte Carlo spectral viability criteria, and a distributed multi-sensor fusion topology using Frobenius norm-based contradiction metrics. Analytical robustness limits are derived, providing deterministic abort thresholds for autonomous systems. Simulation results are presented in tabular form, demonstrating the framework's ability to maintain epistemic safety under temporal decay and adversarial sensor corruption.

In this paper we study the fractional p(•, •)-Laplacian and we introduce the corresponding nonlocal conormal derivative for this operator. We prove basic properties of the corresponding function space and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, we prove the existence of weak solutions of corresponding p(•, •)-Robin boundary problems with sign-changing potentials by applying variational tools.

In this paper we study the fractional p(•, •)-Laplacian and we introduce the corresponding nonlocal conormal derivative for this operator. We prove basic properties of the corresponding function space and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, we prove the existence of weak solutions of corresponding p(•, •)-Robin boundary problems with sign-changing potentials by applying variational tools.

In this paper, we give some properties and remarks of the new fractional Sobolev spaces with variable exponents. We also study the eigenvalue problem involving the new fractional p(•)-Laplacian. Recently, great attention has been focused on elliptic equations involving fractional operators, both for pure mathematical research and in view of concrete real-world applications. This type of operator have major applications to various nonlinear problems, including phase transitions, thin obstacle problem, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes and flame propagation, ultra-relativistic limits of quantum mechanics, multiple scattering, minimal surfaces, material science, water waves, etc. We refer to for a comprehensive introduction to the study of nonlocal problems. In recent years, the study of differential equations and variational problems involving variable exponent conditions has been an interesting topic. Lebesgue spaces with variable exponents appeared in the literature in 1931 in the paper by Orlicz [14]. Zhikov [18] started a new direction of investigation, which created the relationship between spaces with variable exponents and variational integrals with nonstandard growth conditions. For more details on Lebesgue and Sobolev spaces with variable exponents, we refer the reader to . To our best knowledge, Kaufmann et al. firstly introduced some results on fractional Sobolev spaces with variable exponent W s,q(•),p(•,•) (Ω) and the fractional p(•)-Laplacian. There, the authors established compact embedding theorems of these spaces into variable exponent Lebesgue spaces. As an application, they also proved an existence result for nonlocal problems involving the fractional p(•)-Laplacian. In [3], Bahrouni and Rǎdulescu obtained some further qualitative properties of the fractional Sobolev space W s,q(•),p(•,•) (Ω) and the fractional p(•)-Laplacian. After that, some studies on this kind of problems have been performed by using different approaches, see . Let Ω be a bounded Lipschitz domain in R N . For any real s > 0 and for any functions q(x) and p(x, y), we want to define the fractional Sobolev space with variable exponent. We start by fixing s ∈ (0, 1), q ∈ C(Ω, R), and p ∈ C(Ω × Ω, R). Throughout this paper, we assume that 1 < p(x, y) = p(y, x) < N s , ∀ (x, y) ∈ Ω × Ω (P) and 1 < q(x) < Np(x, x) N -sp(x, x) =: p * s (x), ∀ x ∈ Ω.

In this paper, we give some properties of the new fractional Sobolev spaces with variable exponents and apply them to study a class of eigenvalue problems involving the fractional p ( · )-Laplace operator. We obtain sequences of eigenvalues going asymptotically to infinity and we also establish sufficient conditions to get zero value for the principal eigenvalue, which is a striking difference between the variable exponent case and the constant exponent case. As an application, we obtain several existence and nonexistence results for the eigenvalue problem according to the asymptotic growth of the nonlinearity and the range of the spectral parameter.

In this paper, we introduce an iterative procedure for approximating fixed points of a contraction mapping. We also discuss its stability under mild conditions. By exploring the properties related to uniformly convex Banach spaces, we can establish a collection of convergence outcomes, both weak and strong, about generalized nonexpansive mappings. Numerical examples demonstrate that the proposed method outperforms several existing iterative schemes. We validate our main result by applying it to find an approximate solution of a fractional Volterra-Fredholm integro-differential equation. The findings extend the scope of numerous previously published results.

Abstract: - The paper presents a construction theorem for a class of operators dense over the set of causal, time invariant fading memory operators. In this sense, it extends the classical results of S. Boyd and LO Chua that the Volterra series operators are universal approximators for ...

We construct a coercive realization of the Yang-Mills mass gap using Hilbert–de Sitter Spectral Geometry (HdSSG): the operator-theoretic framework built on SO(1,4) principal series representations, de Sitter causal-diamond exhaustion, and Mosco convergence of quadratic forms.
The central result is a canonical equivariant unitary embedding (H⊥, L) → Π_{3/2}(SO(1,4)) via a Poisson-kernel intertwiner W_ξ, where L = Π⊥(-Δ_dS + 2gρ_s)Π⊥ is the HdSSG mass-gap operator. Positivity L > 0 and compact resolvent follow from elliptic theory and Kato-Rellich; the spectral gap Δ_HdSSG > 0 is the Casimir eigenvalue of Π_{3/2}.
A GP condensate Y-junction serves as the coordinate realization, supplying the curvature constants C_A = 0.5060, C_F = 0.7200, C_K = 0.3643 from the normalized vortex ODE analytically. The curvature lock C_K < 1 gives c₁ = 0.6357 > 0.
The mass gap is established through: local surjectivity via ker(P_R*) = {0} (Fredholm adjoint coercion), global injectivity via Mosco limit and spectral positivity, density from spectral truncation, and Weyl-sequence exclusion by the same coercive estimate.
Result: Δ_YM = inf σ(H_YM|_{Ω⊥}) ≥ c₁ · Δ_HdSSG > 0, c₁ = 0.6357. No internal hypotheses remain. Two axioms, no experimentally tuned constants.
Full paper: https://doi.org/10.5281/zenodo.20349116

We report a striking empirical phenomenon in the learned latent geometry of polynomial continued fractions (PCFs): small perturbations to the quadratic coefficient a_2 of the numerator polynomial-an Apéry-sensitive coordinate in the experimental parameterization-produce operator displacements that are 38-127 times smaller than those induced by the elementary index-shift operator, depending on the latent dimension k. We call this effect the Apéry silence. Using the OGRE (Operator Geometry Research Engine) framework and a trained GOTv3 geometric autoencoder, we show that the Apéry perturbation operators T_A+ and T_A-produce near-zero mean displacement vectors, have vanishing or near-vanishing commutators with most tested operators, and behave as epsilon-central elements of the empirical operator algebra at epsilon = 0.01. We interpret the result cautiously: it is evidence for latent flatness in the a_2 direction under the current encoder and perturbation scale, not a proof of an Apéry-specific arithmetic theorem. The paper formulates direct numerical and symbolic tests needed to determine whether this flatness reflects a genuine mathematical invariant of PCF space.

We introduce a systematic empirical study of operator geometry in polynomial continued-fraction (PCF) space using the OGRE (Operator Geometry Research Engine) framework and a GOTv3 geometric autoencoder. On a corpus of 286,310 PCFs with labels measuring proximity to zeta(3), we train a geometric encoder E: PCF-> R^k with a spherically constrained bottleneck and analyze induced operator displacements v_T(x) = E(T(x))-E(x) for a library of ten structurally distinct transformations. The main findings are: (i) the learned PCF latent manifold lies on a near-spherical shell with coefficient of variation below 2% across k=3,...,7; (ii) operators partition into a smooth/disruptive hierarchy, with index-shift transformations producing substantially larger displacements than smooth coefficient transformations; (iii) empirical commutator norms exhibit a block structure separating shift/parity interactions from an approximately central scaling/Apéry block; (iv) Apéry perturbations are geometrically near-invisible relative to other operators; and (v) the smooth/disruptive distinction is robust under latent-dimensional ablation. These results support operator geometry as a quantitative exploratory methodology for mathematical object spaces while leaving formal algebraic interpretation to future symbolic validation.

In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6. Sarason's example 1991

Let p > 1 and let q denote the number such that (1/p) + (1/q) = 1. We give a necessary condition for the product of Toeplitz operators T f T ḡ to be bounded on the weighted Bergman space of the unit ball A p α (α > -1), where f ∈ A p α and g ∈ A q α , as well as a sufficient condition for T f T ḡ to be bounded on A p α . We use techniques different from those in [K. Stroethoff, D. Zheng, Bounded Toeplitz products on Bergman spaces of the unit ball, J. Math. Anal. Appl. 325 (2007) 114-129], in which the case p = 2 was proved.

The Annals of Communications in Mathematics (ISSN 2582-0818, Online) is an international peer-reviewed journal dedicated to publishing high-quality original research papers and invited survey articles in all areas of Pure and Applied Mathematics.

✨ Volume Information (V7N3) (2024)
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Present paper is the study about Stancu type generalization of modified Beta-Szasz operators and their q-analogues. We obtain some approximation properties for these operators and estimate the rate of convergence by using the first and second order modulus of continuity. Author also investigates the statistical approximation properties of the q-Beta-Stancu operators using Korokvin theorem.

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂ t q ±i Sq =0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of t...

In our present investigation, we are concerned with the Kantorovich variant of Lupaş-Stancu operators based on Polya distribution with Pochhammer k-symbol. We briefly give some basic properties of the generalized operators and by making use of these results, we investigate convergence properties of the studied operators. Furthermore, the rate of convergence of these operators is obtained and Voronovskaja type theorem for the pointwise approximation is established. Then we construct bivariate generalization of the operators and we discuss some convergence properties. Finally, taking into account some illustrative graphics, we conclude our study with the comparison of the rate of convergence between our operators and other operators which are mentioned in the paper.

In scalar-valued Hardy space, the class of Schmidt subspaces for a bounded Hankel operator are closely related to nearly S * -invariant subspaces, as described by Gérard and Pushnitski. In this article, we prove that these subspaces in the context of vector-valued Hardy spaces are nearly S * -invariant with finite defect in general. As a consequence, we obtain a short proof of the characterization results concerning the Schmidt subspaces in scalar-valued Hardy space in an alternative way. Thus, our work complements the work of Gérard and Pushnitski regarding the structure of Schmidt subspaces.

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.

In this article, we introduce a new class of conjugations in the scalar valued Hardy space H 2 C (D) and provide a characterization of a complex symmetric Toeplitz operator T φ with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator TΦ on the vector valued Hardy space H 2 C 2 (D) with respect to certain conjugations introduced in .

show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article, we provide a vectorial generalization of a result of Liang and Partington. As an immediate application, we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (see Chattopadhyay et al. in Integr Equ Oper Theory 92(6): 52, 2020, Theorem 3.5 and O'Loughlin in Complex Anal Oper Theory 14(8): 86, 2020, Theorem 3.4) in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space.

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vectorvalued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington. Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.

In this article studies the numerical range of one class of two-parameter spectral problems with compact selfadjoint operators. It is proved that even without the requirement of neither right nor left of definiteness condition, the numerical range of the problem under consideration does not contain some  neighborhood) ;

In this paper, we consider a two-parameter problem with compact, self-adjoint operators in Hilbert spaces. It is proved that if in a weakly coupled system of linear equations, the main tensor-determinant operator takes both positive and negative values, then under the of left definiteness conditions numerical range of this problem consists of the union of two convex closed polygons (finite or infinite). Moreover, the first of these polygons is in the first, and the second of these polygons is in the third quadrant of the real plane. And if the main tensor-determinant operator is a positive (negative) operator, then the numerical image of the problem under the left definiteness condition, consists of only one convex closed polygon (finite or infinite) located in the first (third) quadrant of the real plane.

The Annals of Communications in Mathematics (ISSN 2582-0818, Online) is an international peer-reviewed journal dedicated to publishing high-quality original research papers and invited survey articles in all areas of Pure and Applied Mathematics.

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The Annals of Communications in Mathematics (ISSN 2582-0818, Online) is an international peer-reviewed journal dedicated to publishing high-quality original research papers and invited survey articles in all areas of Pure and Applied Mathematics.

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The Annals of Communications in Mathematics (ISSN 2582-0818, Online) is an international peer-reviewed journal dedicated to publishing high-quality original research papers and invited survey articles in all areas of Pure and Applied Mathematics.

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This document constitutes the closure of the mathematical corpus of the dual-substrate framework, structured around a rigorous formalization of the state of complex systems and their dynamics. The system is defined as : Ω(t) = (G(t), Q(t), I(t), M(t))

The starting point of this work was the announcement by SV Kerov and AM Vershik [11] that the finite characters of the inductive limit group U (oo) can all be obtained as limits of normalized characters of U (N), which we call the Asymptotic Character Formula or the&quot; ergodic method&quot;. In Section 2, we give a detailed proof of this Theorem for representations in the non-negative signatures. It is exactly this case which we need to establish the Asymptotic Formula for the other classical groups: the infinite symplectic, special orthogonal, and spin ...

The starting point of this work was the announcement by SV Kerov and AM Vershik [11] that the finite characters of the inductive limit group U (oo) can all be obtained as limits of normalized characters of U (N), which we call the Asymptotic Character Formula or the&quot; ergodic method&quot;. In Section 2, we give a detailed proof of this Theorem for representations in the non-negative signatures. It is exactly this case which we need to establish the Asymptotic Formula for the other classical groups: the infinite symplectic, special orthogonal, and spin ...

We develop an annulus/Klein-surface model for the completed zeta function ξ(s) on the critical strip and show how the functional equation can be interpreted as a geometric involution on an annulus that descends to a di-analytic map on a Klein surface. Transporting by the exponential change of variables w = exp 2π(s-1 2) , f (w) := ξ 1 2 + (log w)/(2π) , the line Re s = 1 2 corresponds to the unit circle |w| = 1, while Re s = 0 and Re s = 1 map to the boundary circles of an annulus. The functional equation ξ(s) = ξ(1-s) becomes f (w) = f (1/w), which we realize as the angular-reversing gluing of annular boundaries, producing a Klein surface on which f is well defined. On this platform we prove Poisson-Szegő and Jensen-type identities tailored to annuli, and we recast the Riemann Hypothesis as the triviality of the annular Blaschke factor (equivalently, equality of boundary means of log |f |). We complement the analysis with numerical illustrations on the annulus and its Klein-bottle quotient.

This paper develops a finite-window certification layer for Riemann-zero boxes generated from the reduced-residue sieve remainder structure behind the lattice 6k ± 1. It is written as a certification companion to Detecting Riemann zeros from composite-coded sieve remainders in 6k ± 1. The earlier detector paper established the exact zero-carrying object H (6) z (s) = ζ(s)(1-2-s)(1-3-s) 5≤p≤z * AI assistance disclosure: a full statement appears before the references.

This paper assembles the finite-window restricted-Weil operator framework developed in The Geometry of the Critical Line into a single operator-theoretic dictionary. The central object is the constrained minimum m(a) of a finite-window quadratic form q_a on the restricted admissible class. The paper records five live analytic targets, three structural dictionaries, and the named open inputs through which restricted Weil positivity may be studied.
The main structural dictionary identifies the compressed admissible Weil condition (CWC) with truncated Weil zero-side positivity after removal of the trivial-pole rank-one boundary. Equivalently, the lifted form Q+(u) = q_a(u) + 2a |⟨u, cosh(ax/2)⟩|² is the finite-window zero-side Weil form together with the protected odd boundary term. This does not prove the Riemann Hypothesis; it identifies the RH-level residual in the finite-window operator language.
The paper also records the type-balance dictionary governing the β < 3/4 severe-compression route, the bilinear obstruction in the within-window scalar-resolvent continuation problem, the finite-witness certification hierarchy, the retired routes, and the open bridges. The published corpus does not commit to either alternative of the Riemann Hypothesis; it supplies the bidirectional machinery for studying both.

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable
weighted shifts W(훼,훽)
, when the initial data are given as the Berger measure of the restriction of W(훼,훽)
to a canonical invariant subspace, together with the marginal measures for
the 0–th row and 0–th column in the weight diagram for W(훼,훽)
. We prove that the natural
necessary conditions are indeed sufficient. When the initial data correspond to a soluble
problem, we give a concrete formula for the Berger measure of W(훼,훽)
. Our strategy is to
build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of
ROMP for arbitrary canonical invariant subspaces of 퓁2(ℤ2
+).

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable
weighted shifts W(훼,훽)
, when the initial data are given as the Berger measure of the restriction of W(훼,훽)
to a canonical invariant subspace, together with the marginal measures for
the 0–th row and 0–th column in the weight diagram for W(훼,훽)
. We prove that the natural
necessary conditions are indeed sufficient. When the initial data correspond to a soluble
problem, we give a concrete formula for the Berger measure of W(훼,훽)
. Our strategy is to
build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of
ROMP for arbitrary canonical invariant subspaces of 퓁2(ℤ2
+).

We consider the one-dimensional potential scattering problem on the line when the potential is given by a self-similar fractal measure. We show that the scattering operator admits a Weyl-Berry like expansion in terms of operators, where the second order term has a non trivial scaling which re ects the fractal nature of the potential.