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Advances in Operator Theory and Nonlinear Evolution Equations

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Dr. Yoritaka Iwata

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Faculty of International Studies, Osaka University of Economics and Law, Yao 581-0853, Japan
Interests: operator algebra; functional analysis; nonlinear semigroup

Special Issue Information

Dear Colleagues,

Recent progress in abstract operator theory is shown with respect to real and complex analysis. Although discussion is basically developed in the abstract Banach spaces framework, more concrete mathematical spaces such as Besov spaces are employed in some cases.

Among others, much attention is paid on the abstract operator theory applied to the nonlinear evolution equations, where the formation of certain kinds of nonlinear semigroups and its profiling is one of the key issues. Consequently, from an operator algebraic point of view, the sub- and super-algebraic structures of Banach algebra is found. For accomplishing the classification of generally unbounded operator, the integral representations of operators in the complex plane are effectively used, so that those concepts are explained.

Dr. Yoritaka Iwata
Guest Editor

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Published Papers (3 papers)

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Research

9 pages, 244 KB

Open AccessArticle

Equality Between the Spectrum of PT-Symmetric Shrödinger Operators and Their Adjoint

by Ece Özdemir and Alp Arslan Kıraç

Mathematics 2026, 14(4), 608; https://doi.org/10.3390/math14040608 - 10 Feb 2026

Viewed by 301

Abstract

In classical spectral theory, self-adjoint differential operators satisfy the relation

σ (L) = σ (L^{*})

. However, this is not necessarily true for non-self-adjoint operators. In this study, we show that a similar spectral equality holds for non-self-adjoint [...] Read more.

In classical spectral theory, self-adjoint differential operators satisfy the relation

σ (L) = σ (L^{*})

. However, this is not necessarily true for non-self-adjoint operators. In this study, we show that a similar spectral equality holds for non-self-adjoint with periodic PT-symmetric complex valued potential. These operators are often described in the literature as having self-adjoint-like spectral characteristics; however, here, we show that the reality of the spectrum depends on the isospectral relationship between the operator and its adjoint. Moreover, we prove that the adjoint operator

L^{*}

inherits PT-symmetry and analyze its spectral properties under quasi-periodic boundary conditions, extending prior studies from the original operator to its adjoint. Full article

(This article belongs to the Special Issue Advances in Operator Theory and Nonlinear Evolution Equations)

13 pages, 292 KB

Open AccessArticle

Pseudo-Almost Automorphic 𝓒⁰-Solutions: Well-Posedness and Asymptotic Behaviour in Evolution Equations with Nonlocal Constraints

by Bassem Meknani, Reem Alrebdi and Ahmed Bchatnia

Mathematics 2026, 14(3), 459; https://doi.org/10.3390/math14030459 - 28 Jan 2026

Viewed by 274

Abstract

This research establishes an innovative analytical framework for examining

C^{0}

-solutions of pseudo-almost automorphic type in evolution systems governed by nonlocal initial conditions within Banach spaces. Our methodological approach is founded upon the compactness properties of the resolvent operator [...] Read more.

This research establishes an innovative analytical framework for examining

C^{0}

{(I - A)}^{- 1}

and the semigroup generated by A. By integrating semigroup theory with fixed-point methodologies, we develop a unified strategy that successfully overcomes challenges arising from nonlocal initial specifications. The practical applicability of our theoretical framework is verified through its implementation on a transport equation with nonlocal initial history, thereby demonstrating both the existence of solutions and their distinctive character. Full article

(This article belongs to the Special Issue Advances in Operator Theory and Nonlinear Evolution Equations)

13 pages, 318 KB

Open AccessArticle

Weighted Approximation by Szász–Mirakyan–Durrmeyer Operators Reproducing Exponential Functions

by Gülsüm Ulusoy Ada and Ali Aral

Mathematics 2026, 14(1), 59; https://doi.org/10.3390/math14010059 - 24 Dec 2025

Cited by 1 | Viewed by 511

Abstract

We examine a Szász–Mirakyan–Durrmeyer type operator that reproduces the functions 1 and

e^{2 a x}

for a fixed parameter

a > 0 .

While its exponential reproduction property has been described in the classical literature, the effect of exponential weights on its [...] Read more.

We examine a Szász–Mirakyan–Durrmeyer type operator that reproduces the functions 1 and

e^{2 a x}

for a fixed parameter

a > 0 .

While its exponential reproduction property has been described in the classical literature, the effect of exponential weights on its approximation behavior has not been studied. In this work, we provide a detailed analysis of the operator in weighted spaces and show that combining exponential reproduction with weighted norms improves the approximation behavior for exponentially growing functions. We also prove that the corresponding sequence of operator norms remains uniformly bounded for a family of exponential weights, ensuring the stability of the operators in the weighted framework. Moreover, we establish new Korovkin-type approximation theorems involving weighted convergence and obtain sharp uniform error estimates in the presence of exponential weights. These results extend the classical theory to weighted exponential settings and highlight several quantitative features that do not arise in the classical case. Full article

(This article belongs to the Special Issue Advances in Operator Theory and Nonlinear Evolution Equations)

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