Mehmet Gurdal

Süleyman Demirel University | SDU · Department of Mathematics

This book presents contemporary mathematical concepts and techniques including theories of summability, fixed point and non-absolute integration and applications, providing an overview of recent developments in the foundations of the field as well as its applications. It discusses the recent results of double sequence spaces as the four-dimensional...

This paper explores the concepts of J-lacunary statistical limit points, J-lacunary statistical cluster points, and J-lacunary statistical Cauchy multiset sequences. Building upon previous work in the field, we investigate the relationships between J-lacunary statistical convergence and J *-lacunary statistical convergence in multiset sequences. Th...

This paper introduces the neutrosophic -statistical convergent difference sequence spaces defined through a modulus function. Additionally, we establish new topological spaces and examine various topological properties within these neutrosophic -statistical convergent difference sequence spaces.

The primary objective of this study is to introduce the notion of ideal convergence in quaternion-valued generalized metric spaces. We define Iconvergence and I ∗-convergence in these spaces and establish their equivalence through the definition of property (AP). Furthermore, we introduce I-Cauchy and I ∗-Cauchy sequences, adapting classically theo...

In this paper, we establish upper bounds that both extend and enhance known inequalities for the Berezin number of bounded linear operators and product of operators. In particular, this paper aims to establish new upper bounds on the Berezin number for operators on functional Hilbert spaces by introducing important improvements to the Buzano inequa...

This study focuses on developing and examining new definitions related to statistical convergence, modulus functions, and σ-density in 2-normed spaces. We explore various relationships between statistical f_σ-convergence, f_σ-statistical convergence, f_σ-statistically Cauchy sequences, f_σ-statistically bounded double sequences, and strongly f_(σ_s...

Motivated by the previously reported results, this work attempts to provide fresh refinements to both vector and numerical radius inequalities by providing a refinement to the well known Buzano's inequality which as a consequence yielded another refinement of the Cauchy-Schwartz (CS) inequality. Utilizing the new refinements of the Buzano's and Cau...

This work presents applications of certain functions to Berezin radius inequalities, as well as further proofs of inequalities linking them. We also establish new inequalities using the operator on a functional Hilbert space concerning the Berezin radius and Berezin norm of specific finite sums. There are also several Berezin radius inequalities pr...

This paper defines the space S_(θ_uv)^α (Δ_s^j,f), encompassing all sequences that are (Δ_s^j,f)-lacunary statistically convergent of order α, utilizing an unbounded modulus function f, a double lacunary sequence θ_uv={(k_u,l_v )}, a generalized difference operator Δ_s^j, and a real number α ∈ (0,1]. Additionally, the space ω_(θ_uv)^α (Δ_s^j,f) is...

The primary objective of this study is to introduce the concepts of $I_2$-deferred Cesàro summability and $I_2-$ deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of $I_2$-d...

We compute certain inequalities for B-Berezin radius of \(2\times 2\) operator matrices in the study that generalize and refine earlier inequalities. Furthermore, we construct A-Berezin radius inequalities of operators in \(\mathbb {B}_{A,\Upsilon }(\mathcal {H})\) that improve on the current inequalities in Huban (Turk J Math 46(1):189–206, 2022)....

Introduction/purpose: In this article, the author's goal is to seek to obtain new inequalities of the Berezin type. Methods: The methods used are standard for operator theory. Results: Various inequalities of the type given by Huban et al. and Erkan have been obtained. Conclusions: In addition to obtaining various inequalities of the form given by...

The purpose of this research is to show bounds for some Berezin number inequalities in an innovative approach. Some inequalities have been proven using the improvement of the Hermite-Hadamard inequality. These inequalities are a refined version of Huban et al.'s inequalities (Huban et al., 2021b; Huban et al., 2022a) and Başaran et al.'s inequaliti...

Our main goal in this paper is to introduce the concept of ideal convergence in G-metric spaces. We give definitions of GI-convergence and GI*-convergence in G-metric spaces. We also extend the I-convergence concept's properties to GI-convergence. Then we demonstrate that GI-convergence and GI*-convergence are equivalent by giving the property (AP)...

In this manuscript, we introduce the concepts of strong N h p ð Þ-summability of order a and lacunary statistical convergence of order a for fuzzy variables in credibility space. We examine important connections between these ideas. The circumstances of lacunary statistical convergence almost surely (a.s.) of order a, lacunary statistical convergen...

In this study, we investigate the notions of ℐ 2-convergence almost surely (a.s.) and ℐ2-convergence a.s. of complex uncertain double sequences in an uncertainty space, and obtain some of their features and identify the relationships between them. In addition , we put forward the concepts of ℐ 2 and ℐ2-Cauchy sequence a.s. of complex uncertain doub...

In this paper, the S θ (∆) and N θ (∆) summabilities are used along with the notion of weakly unconditionally Cauchy series (in brief wuC series) to characterize a Banach space. We examine these two kinds of summabilities which are regular methods and we recall some features. Furthermore, we investigate the spaces S N θ (p ∆w p) and S S θ (p ∆w p)...

In this research, we investigate multiple conceptions of convergence and deferred statistical convergence of order \(\beta ,\) \(\left(0<\beta \le 1\right)\) for fuzzy variable sequences within framework credibility theory. The idea of deferred statistical convergence of order \(\beta\) for fuzzy variable sequences such as the notions of convergenc...

In this manuscript, we introduce the concepts of strong \(N_{\theta }\left( p\right)\)-summability of order \(\alpha\) and lacunary statistical convergence of order \(\alpha\) for fuzzy variables in credibility space. We examine important connections between these ideas. The circumstances of lacunary statistical convergence almost surely (a.s.) of...

For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{...

İşlevsel Hilbert uzayları, istatistik, yaklaşım teorisi, grup temsili teorisi, vb. dahil olmak üzere birçok alanda ortaya çıkar. İşlevsel Hilbert uzay sayesinde tanımlanan Berezin dönüşümü ise, düzgün fonksiyonları analitik fonksiyonların Hilbert uzayları üzerindeki operatörlerle ilişkilerini inceler. Berezin yarıçapını ve Berezin normunu karakteri...

– In functional analysis, linear operators induced by functions are frequently encountered; thesecontain Hankel operators, constitution operators, and Toeplitz operators. The symbol of the resultantoperator is another name for the inciting function. In many instances, a linear operator on a Hilbert spaceℋ results in a function on a subset of a topo...

We investigate new upper bounds for the Berezin radius and Berezin norm of $2\times2$ operator matrices using the Cauchy-Buzano inequality, and we propose a required condition for the equality case in the triangle inequalities for the Berezin norms. We also show various Berezin radius inequalities for matrices with $2\times2$ operators.

Dear Professor Greetings!! We are glad to inform you that an edited book entitled " Fixed point, Summability theory and non absolute integrals" is under process for publication with Springer Nature. This edited book will be indexed in Scopus. Thus we are inviting you to contribute your original research articles to this edited book. Best regards...

In this paper, we put forward rough statistical φ-convergence of difference sequences as a generalization of rough statistical convergence as well as statistical φ-convergence. We study some of its fundamental properties. We obtain some results for rough statistical φ-convergence for difference double sequences by introducing the rough statistical-...

In this paper, we have introduced the notion of the lacunary [Formula: see text]-statistical convergence of triple sequences for rough variables. In addition, we have defined lacunary [Formula: see text]-statistical Cauchy sequence of rough variables in trust space and given the lacunary [Formula: see text]-statistical completeness for trust space....

Fast [12] is credited with pioneering the field of statistical convergence. This topic has been researched in many spaces such as topological spaces, cone metric spaces, and so on (see, for example [19, 21]). A cone metric space was proposed by Huang and Zhang [17]. The primary distinction between a cone metric and a metric is that a cone metric is...

In the present article, we introduce the concepts of strongly asymptotically lacunary equivalence, asymptotically statistical equivalence, asymptotically lacunary statistical equivalence for sequences in gmetric spaces. We investigate some properties and relationships among this new concepts.

The aim of this article is to investigate the neutrosophic Nörlund-statistically convergent sequence space. We present some neutrosophic normed spaces (NNSs) in Nörlund convergent spaces. In addition, we also examine various topological and algebraic properties of these convergent sequence spaces. Theorems are proved in light of the NNS theory appr...

The aim of this paper is to investigate the intuitionistic Nörlund [Formula: see text]-statistically convergent sequence space. We present some intuitionistic fuzzy normed spaces (IFNS) in Nörlund convergent spaces. Moreover, we also put forward several topological and algebraic properties of these convergent sequence spaces.

We investigate the rough statistical convergence of complex uncertain triple sequences in this research. We show three forms of rough statistically convergent complex uncertain triple sequences and rough lambda3-statistical convergence in measure, as well as other fundamental features.

In the present article, we introduce the concepts of strongly asymptotically lacunary equivalence, asymptotically statistical equivalence , and asymptotically lacunary statistical equivalence for sequences in g-metric spaces. We investigate some properties and relationships among these new concepts.

We consider the Riccati operator equations on the weighted Bergman space A2? (Bn) of the unit ball Bn in Cn and investigate the properties of their solutions. Our discussion uses the Berezin symbols method.

The main aim of this investigation is to introduce rough I-statistical convergence in probabilistic n-normed spaces (briefly Pr-n-spaces). We establish some results on roughI-statistical convergence and also we introduce the notion of rough I-statistical limit set in Pr-n-spaces and discuss some topological aspects on this set. Moreover, we define...

In this paper, we present the ideal convergence of triple sequences for rough variables. Furthermore, sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper presents two types of ideal convergence of rough triple sequence: Convergence in trust and convergence in mean. Some mathematical properties...

We consider the space \(C^{\left( n\right) }\left( \Omega \right) ,\) the Banach space of continuous functions with n derivatives and the n th derivative continuous in \({\overline{\Omega }},\) where \(\Omega \subset {\mathbb {C}}\) is a starlike region with respect to \(\alpha \in \Omega .\) We use the so-called \(\alpha\)-Duhamel product $$\begin...

In this paper, we present the notions of lacunary statistically convergent sequence for fuzzy variables, lacunary statistically Cauchy sequence in credibility space, and present a kind of lacunary statistical completeness for credibility space. Also, we present lacunary strong convergence concepts of sequences of fuzzy variables of different types.

In his paper, within frame work credibility theory, we examine several notions of convergence and statistical convergence of fuzzy variable sequences. The convergence of fuzzy variable sequences such i is the notion of convergence incredibility, convergence in distribution, convergence in mean, and convergence uniformly virtually certainly via post...

The intent of this paper is to investigate the intuitionistic Nörlund [Formula: see text]-lacunary statistically convergent sequence space. We present some intuitionistic fuzzy normed spaces (IFNS) in Nörlund convergent spaces. Moreover, we also put forward several topological and algebraic properties of these convergent sequence spaces.

The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $Q$ with normalized reproducing kernel $k_{\eta}:=\dfrac{K_{\eta}}{\left\Vert K_{\eta}\right\Vert}$ are defined, respectively, by $\widetilde{A}(\eta)=\left\langle {A}k_{\eta},k_{\eta}\right\rangle$, $\eta\in Q$ and...

Smooth functions are associated with operators on Hilbert spaces of analytic functions through the Berezin transform. The Berezin symbol and the Berezin number of an operator A on the Hilbert functional space H(Ω) over some set Ω with the reproducing kernel are defined, respectively, by A ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω and ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|...

Let A be a positive bounded linear operator acting on a complex Hilbert space H. Let ber A (X) denote the A-Berezin number of an operator X. In this paper, we give new inequalities of A-Berezin number of operators on the reproducing kernel Hilbert space. Some more related results are also obtained. In particular, we show that ber n A (X) ≤ 1 2 n−1...

In the present paper we introduce and study Orlicz lacunary convergent triple sequences over n-normed spaces. We make an effort to present the notion of $g_{3}$-ideal convergence in triple sequence spaces. We examine some topological and algebraic features of new formed sequence spaces. Some inclusion relations are obtained in this paper. Finally,...

In this paper, we present the notions of statistically convergent sequence for fuzzy variables, statistically Cauchy sequence in credibility space, and present a kind of statistical completeness for credibility space. Furthermore, the conditions of statistical convergence almost surely (a.s.), statistical convergence in credibility, statistical con...

We introduce the concepts statistical cluster and statistical limit points of a sequence of fuzzy numbers in a fuzzy valued metric space. Then we obtain some inclusion relations between the sets of limit points, statistical limit points and statistical cluster points for a sequence of fuzzy numbers.

In this article we introduce some new type of summability methods for double sequences involving the ideas of de la Vallée-Poussin mean in probabilistic 2 -normed space and examine some important results.

The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator $T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $ over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined, respectively, by \[ \widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta }\right\Vert }},\frac{K_{\eta}}{{\left\...

The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $\Omega$ with normalized reproducing kernel $\widehat{k}_{\lambda}$ are defined, respectively, by $\widetilde{A}(\lambda)=\left\langle {A}\widehat{k}_{\lambda },\widehat{k}_{\lambda}\right\rangle ,\ \lambda\in\Omega$...

Let ? be a fixed complex number, and let ? be a simply connected region in complex plane C that is starlike with respect to ? ? ?. We define some Banach space of analytic functions on ? and prove that it is a Banach algebra with respect to the ?-Duhamel product defined by (f?? g)(z) := d/dz z?? f(z+??t)g(t)dt. We prove that its maximal ideal space...

The Berezin symbol ?A of an operator A on the reproducing kernel Hilbert space H (?) over some set ? with the reproducing kernel k? is defined by ? (?) = ?A k?/||k?||, k?/||k?||?, ? ? ?. The Berezin number of an operator A is defined by ber(A) := sup ??? |?(?)|. We study some problems of operator theory by using this bounded function ?, including t...

Normalleştirilmiş $K_{\lambda}:=\frac{k_{\lambda}}{\left\Vert k_{\lambda}\right\Vert_{\mathcal{H}}}$, üretici çekirdekli $\mathcal{H}\left( \Omega\right) $, Hilbert uzayı üzerinde $A$ sınırlı lineer operatör için Berezin sembolü ve Berezin sayısı sırasıyla $A\left( \lambda\right) :=\left\langle AK_{\lambda},K_{\lambda}\right\rangle _{\mathcal{H}}$...

In this paper, some existing theories on convergence of fuzzy number sequences are extended to I2-statistical convergence of fuzzy number sequence. Also, we broaden the notions of I-statistical limit points and I-statistical cluster points of a sequence of fuzzy numbers to I2-statistical limit points and I2-statistical cluster points of a double se...

In this paper, our concern is to introduce the concepts of invariant arithmetic convergence, invariant arithmetic statistically convergence and lacunary invariant arithmetic statistically convergence using weighted density via Orlicz function φe. Finally, we give some relations between lacunary invariant arithmetic statistical φe-convergence and in...

In this paper, our concern is to introduce the concepts of invariant arithmetic convergence, invariant arithmetic statistically convergence and lacunary invariant arithmetic statistically convergence using weighted density via Orlicz function φe. Finally, we give some relations between lacunary invariant arithmetic statistical φe-convergence and in...

In this work, we study the lacunary I -statistical convergence concept of complex uncertain triple sequence. Four types of lacunary I -statistically convergent complex uncertain triple sequences are presented, namely lacunary I -statistical convergence in measure, in mean, in distribution and with respect to almost surely, and some basic properties...

In this research paper, we analyze the lacunary statistical convergence and lacunary statistical Cauchy concepts of triple sequence in fuzzy metric space. We also introduce the concept of triple lacunary statistical completeness and prove some basic properties.

In this paper, we define analogies of classical H?lder-McCarthy and Young type inequalities in terms of the Berezin symbols of operators on a reproducing kernel Hilbert space H = H (?). These inequalities are applied in proving of some new inequalities for the Berezin number of operators. We also define quasi-paranormal and absolute-k-quasi paranor...

In this work, we construct the transformation operator for the infinite system of the difference equations $a_{n-2}y_{n-2}+b_{n-1}y_{n-1}+c_{n}y_{n}+b_{n}y_{n+1}+a_{n}y_{n+2}=\lambda y_{n}$ $(n=1,2,...)$,where $a_{n}\neq0,$ $b_{n},$ $c_{n}$ $(n=1,2,3,...)$ are given complex numbers, investigate some important properties of the special solutions of...

In this paper we have introduced the I-localized and the I^{∗}-localized sequences in metric spaces and investigate some basics properties of the I-localized sequences related with I-Cauchy sequences. Also we have obtained some necessary and sufficient conditions for the I-localized sequences to be an I-Cauchy sequences. It is also defined uniforml...

In this article, we use Kantorovich and Kantorovich type inequalities in order to prove some new Berezin number inequalities. Also, by using a refinement of the classical Schwarz inequality, we prove Berezin number inequalities for powers of f (A), where A is self-adjoint operator on the Hardy space H 2(D) and f is a positive continuous function. S...

We introduce statistically localized sequences in 2-normed spaces and give some main properties of statistically localized sequences. Also, we prove that a sequence is statistically Cauchy sequence if and only if its statistical barrier is equal to zero. Moreover, we define the uniformly statistically localized sequences on 2-normed spaces and inve...

In 1974, Krivonosov defined the concept of localized sequence that is defined as a generalization of Cauchy sequence in metric spaces. In this work, by using the concept of ideal, the statistically localized sequences are defined and some basic properties of \(\mathcal {I}\)-statistically localized sequences are given. Also, it is shown that a sequ...

A reproducing kernel Hilbert space (shorty, RKHS) H=H(Ω) on some set Ω is a Hilbert space of complex valued functions on Ω such that for every λ∈Ω the linear functional (evaluation functional) f→f(λ) is bounded on H. If H is RKHS on a set Ω, then, by the classical Riesz representation theorem for every λ∈Ω there is a unique element kH,λ∈H such that...

In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition operators on H2 are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the...

In this paper, we study the Berezin number inequalities by using the transform \(C_{\alpha ,\beta }\left( A\right) \) on reproducing kernel Hilbert spaces (RKHS). Moreover, we give Grüss-type inequalities for selfadjoint operators in RKHS.

We prove analogs of certain operator inequalities, including Hölder-McCarthy inequality, Kantorovich inequality, and Heinz-Kato inequality, for positive operators on the Hilbert space in terms of the Berezin symbols and the Berezin number of operators on the reproducing kernel Hilbert space.

In this paper we have introduced the statistically localized sequences in metric spaces and investigate basic properties of the statistically localized sequences. Also we have obtained some necessary and sufficient conditions for a localized sequence to be a statistically Cauchy sequence. It is also defined uniformly statistically localized sequenc...

In this article, we are interested in the zero Toeplitz product problem: for two symbols f; g ∈ L ∞ (D(double-struck); dA), if the product TfTg is identically zero on Lα² (D(double-struck)) ; then can we claim Tf or Tg is identically zero? We give a particular solution of this problem. A new proof of one particular case of the zero Toeplitz product...

In this paper, we introduce a new type of convergence for a sequence of function, namely, \(\lambda \)-statistically convergent sequences of functions in random 2-normed space, which is a natural generalization of convergence in random 2-normed space. In particular, following the line of recent work of Karakaya et al. [12], we introduce the concept...

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space Kθ = H2ΘθH2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated To...

Let A be a Banach algebra with a unit e, and let a ∈ A be an invertible element. We define the following algebra: (forumala presented). In this article we study some properties of this algebra; in particular, we prove that B loce+p = {x ∈ A: px (e − p) = 0}, where p is an idempotent in A. We also investigate the following Deddens subspace. Let a, b...

A Hardy type inequality for Reproducing Kernel Hilbert Space operators is proved. It is well known (see Halmos in A Hilbert space problem book. Springer, Berlin, 1982) the following power inequality for numerical radius of Hilbert space operator A: $$\begin{aligned} w\left( A^{n}\right) \le \left( w\left( A\right) \right) ^{n} \end{aligned}$$for an...

The present paper studies uniqueness properties of the solution of the inverse problem for the Sturm-Liouville equation with discontinuous leading coefficient and the separated boundary conditions. It is proved that the considered boundary-value is uniquely reconstructed, i.e. the potential function of the equation and the constants in the boundary...

The fundamental inequality w (Aⁿ) ≤ wⁿ(A); (n = 1, 2, …) for the numerical radius is much studied in the literature. But the inverse inequalities for the numerical radius are not well known. By using Hardy-Hilbert type inequalities, we give inverse numerical radius inequalities for reproducing kernel Hilbert spaces. Also, we obtain inverse power in...

Following the line of the recent work by Savaş, et al., we apply the notion of ideals to A-statistical cluster points. We get necessary conditions for two matrices to be equivalent in a sense of AI-statistical convergence. In addition, we use Kolk’s idea to define and study BI-statistical convergence.

We prove in terms of so-called Berezin symbols some theorems for Borel summability method for sequences and series of complex numbers. Namely, we characterize the Borel convergent sequences and series; prove regularity of Borel summability method, and prove a new Tauberian type theorem for Borel summability.

In this study we derive the Gelfand-Levitan-Marchenko type main integral equation of the inverse problem for the boundary value problem L and prove the uniquely solvability of the main integral equation. Further, we give the solution of the inverse problem by the spectral data and by two spectrum.

By using Hardy-Hilbert’s inequality, some power inequalities for the Berezin number of a self-adjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs) with applications for convex functions are given.

We give operator analogues of some classical inequalities, including Hardy and Hardy-Hilbert type inequalities for numbers. We apply these operator forms of such inequalities for proving some power inequalities for the so-called Berezin number of self-Adjoint and positive operators acting on Reproducing Kernel Hilbert Spaces (RKHSs). More precisely...

We give an application of Berezin symbols technique in Abel convergence of some sequences and series of complex numbers. Namely, in terms of the growths of Berezin symbols of associated weighted shift operators on the weighted Bergman space on the unit disc D of the complex plane C, we characterize Abel convergence of some sequences and series. We...

In this paper, we introduce the notion of discrete statistical Borel convergence. Also, we give necessary and sufficient condition under which a series with bounded sequence of complex numbers is discrete statistically Borel convergent. Moreover, we present in terms of Berezin symbols some characterization Schatten-von Neumann class operators.

We introduce the notion of Engliš algebras, defined in terms of reproducing kernels and Berezin symbols. Such algebras were apparently first investigated by Engliš (1995). Here we give some new results on Engliš C∗-algebras on abstract reproducing kernel Hilbert spaces and some applications to various questions of operator theory. In particular, we...