arXiv subjects

Adem Kılıçman

Publications and source records attributed to Adem Kılıçman.

10 recordsLinked to original sources

arXiv2021

A Systematic Review on Hermite-Hadamard Inequality: Theory and Applications

Inequalities play important roles not only in mathematics, but also in other fields, such as economics and engineering. Even though many results are published on Hermite-Hadamard (H-H) type inequalities, new researcher to this fields often found it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H type inequalities via various classes of convexity, through differentiable mappings and for fractional integrals, are presented. Some well-known examples from previous literature are used as illustrations.

math.CA↗

arXiv2016

On the direct sum of two bounded linear operators and subspace-hypercyclicity

In this paper, we show that if the direct sum of two operators is subspace-hypercyclic (satisfies subspace hypercyclic criterion), then both operators are subspace-hypercyclic (satisfy subspace hypercyclic criterion). Moreover, if an operator $T$ satisfies subspace-hypercyclic criterion, then so $T\oplus T$ does. Also, we obtain that under certain conditions, if $T\oplus T$ is hypercyclic then $T$ satisfies subspace-hypercyclic criterion and, the subspace-hypercyclic operators satisfy subspace-hypercyclic criterion which gives the "subspace-hypercyclic" analogue of Theorem 2.3. (in Hereditarily hypercyclic operators, J. Funct. Anal., 167:94--112, 1999 by P. Bés and A. Peris).

math.FA↗

arXiv2016

Some new $I_λ$-lacunary statistical convergence

In this paper, we introduce some new $I_λ$-lacunary statistically convergent sequence spaces of order $α$ defined by a Musielak-Orlicz function. We study some relations between $I_λ$-lacunary statistically convergence with lacunary $I_λ$-summable sequences. Moreover we also study about the $I_λ$-lacunary statistically convergence of sequences in topological groups and give some important inclusion theorems.

math.FA↗

arXiv2015

On subspaces diskcyclicity

In this paper, we define and study subspace-diskcyclic operators. We show that subspace-diskcyclicity does not imply to diskcyclicity. We establish a subspace-diskcyclic criterion and use it to find a subspace-diskcyclic operator that is not subspace-hypercyclic for any subspaces. Also, we show that the inverse of invertible subspace-diskcyclic operators do not need to be subspace-diskcyclic for any subspaces. Finally, we prove that every finite-dimensional separable Hilbert space over the complex field supports a subspace-diskcyclic operator.

math.FA↗

arXiv2015

k-bitransitive and compound operators on Banach spaces

In this this paper, we introduce new classes of operators in complex Banach spaces, which we call k-bitransitive operators and compound operators to study the direct sum of diskcyclic operators. We create a set of sufficient conditions for k-bitransitivity and compound. We show the relation between topologically mixing operators and compound operators. Also, we extend the Godefroy-Shapiro Criterion for topologically mixing operators to compound operators.

math.FA↗

arXiv2015

A review of some works in the theory of diskcyclic operators

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if $x\in {\mathcal H}$ has a disk orbit under $T$ that is somewhere dense in ${\mathcal H}$ then the disk orbit of $x$ under $T$ need not be everywhere dense in ${\mathcal H}$. We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space ${\mathcal H}$ over the field of complex numbers if and only if $\dim({\mathcal H})=1$ or $\dim({\mathcal H})=\infty$ . Finally we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

math.FA↗

arXiv2015

Subspace-hypercyclic weighted shifts

Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces $\ell^{2}(\mathbb Z)$ and $\ell^{2}(\mathbb N)$ to be subspace-transitive, consequently, we show that the Herrero question (D. A. Herrero. Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991)179-190) holds true even on a subspace of a Hilbert space, i.e. there exists an operator $T$ such that both $T$ and $T^*$ are subspace-hypercyclic operators for some subspaces. We display the conditions on the direct sum of two invertable bilateral forward weighted shift operators to be subspace-hypercyclic.

math.FA↗

arXiv2015

Operators with Diskcyclic Vectors Subspaces

In this paper, we prove that if $T$ is diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of $T$ is dense in $\mathcal H$. Also, if $T$ is diskcyclic operator and $|λ|\le 1$, then $T-λI$ has dense range. Moreover, we prove that if $α>1$, then $\frac{1}αT$ is hypercyclic in a separable Hilbert space $\mathcal H$ if and only if $T \oplus αI_{\mathbb{C}}$ is diskcyclic in $\mathcal H \oplus \mathbb{C}$. We show at least in some cases a diskcyclic operator has an invariant, dense linear subspace or an infinite dimensional closed linear subspace, whose non-zero elements are diskcyclic vectors. However, we give some counterexamples to show that not always a diskcyclic operator has such a subspace.

math.FA↗

arXiv2014

Some properties of subspaces-hypercyclic operators

In this paper, we answer a question posed in the introduction of \cite{sub hyp} positively, i.e, we show that if $T$ is $\mathcal M$-hypercyclic operator with $\mathcal M$-hypercyclic vector $x$ in a Hilbert space $\mathcal H$, then $P(Orb(T,x))$ is dense in the subspace $\mathcal M$ where $P$ is the orthogonal projection onto $\mathcal M$. Furthermore, we give some relations between ${\mathcal M}^{\perp}$-hypercyclicity and the orthogonal projection onto ${\mathcal M}^{\perp}$. We also give sufficient conditions for a bilateral weighted shift operators on a Hilbert space $\ell^{2}(\mathbb Z)$ to be subspace-hypercyclic, cosequently, there exists an operator $T$ such that both $T$ and $T^*$ are subspace-hypercyclic operators. Finally, we give an $\mathcal M$-hypercyclic criterion for an operator $T$ in terms of its eigenvalues.

math.FA↗

arXiv2010

Note on the Calculation of Groebner-Shirshov Bases for Affine Weyl Groups

In this work we will consider the calculation of Groebner-Shirshov bases of Coxeter groups. This will be the main focus of the work. In \cite{Bokut-Shiao}, Bokut & Shiao gave the Groebner-Shirshov bases of positive definite classical Coxeter groups $A_l, B_l, D_l$ by using the techniques of Elimination of Leading Word. We will give a counter example to a hypothesis which is introduced by Bokut & Shiao in \cite{Bokut-Shiao} and we will calculate the Groebner-Shirshov bases of the positive degenerate infinite affine Weyl group $\widetilde{A}_n $ which is isomorphic to semi-direct product group $Σ_n \ltimes {\mathbb Z}^{n-1}$, and further we classify all the reduced elements of the group by using the Composition Diamond Lemma.

math.AT↗