Ghadir Sadeghi

• Professor
• Hakim Sabzevari University

Projective Hilbert spaces as the underlying spaces of this paper are obtained by identifying two vectors of a Hilbert space $\mathcal{H}$ which have the same phase and denoted by $\hat{\mathcal{H}}$. For a family $\Phi$ of vectors of $\mathcal{H}$ we introduce a topology $\tau_{\Phi}$ on $\hat{\mathcal{H}}$ and provide a topology-based approach for...

Suppose that \({\mathfrak {N}}\) is a von Neumann algebra and that \(\varphi \) is a normal semifinite trace on \({\mathfrak {N}}\). We study bands, projection bands, and band projections in the setting of noncommutative Banach function spaces. Let \(S({\mathfrak {N}}, \varphi )\) be the set of all \(\varphi \)-measurable operators. A subspace \({\...

We generalize some maximal probability inequalities, proven for a class of random variables, to the measure-free setting of Riesz spaces. We prove generalizations of the Kolmogorov inequality, Hájek–Rényi inequality, Lévy’s inequality and Etemadi’s inequality.

We introduce some measures of the dependence such as the strong mixing and uniform mixing coefficients in von Neumann algebras and then define the noncommutative strong and uniform mixing sequences. We establish some notable nonncommutative mixing inequalities such as Ibragimov inequality. Moreover, we extend the notion of mixingale sequence to the...

A new class of the Itô integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is a motivitation to construct near-martingales in Riesz spaces. Furthermore, we state Doob–Meyer decomposition theorem for near-submartingales in Riesz sp...

In this article we consider the m-topology on M (X, A), the ring of all real measurable functions on a measurable space (X, A), and we denote it by M m (X, A). We show that M m (X, A) is a Hausdorff regular topological ring, moreover we prove that if (X, A) is a T-measurable space and X is a finite set with |X| = n, then M m (X, A) ∼ = R n as topol...

In this paper, we define an orthonormal basis for 2-*-inner product space and obtain some useful results. Moreover, we introduce a 2-norm on a dense subset of a 2- * -inner product space. Finally, we obtain a version of the Selberg, Buzano’s and Bessel inequality and its results in an A -2-inner product space.

In this paper, we introduce the concept of K-fusion frames and propose the duality for such frames. The relation between the local frames of K-fusion frames with their duals is studied. The elements from the range of a bounded linear operator K can be reconstructed by K-frames. Also, we establish K-fusion frame multipliers and investigate reconstru...

We introduce the notion of a tree martingale in a noncommutative probability space and prove the Burkholder–Gundy inequalities for tree martingales in symmetric operator spaces. In particular, we establish some inequalities in this setting via an approach based on the concept of generalized singular valued functions of noncommutative random valuabl...

Finding a best dual frame that minimizes the reconstruction errors when erasures occur is a deep-rooted problem in frame theory. The primary purpose of this paper is to introduce a new measurement for constructing optimal duals. We consider the numerical radius of the error operator which has some advantages over the previous measures. Then we give...

We introduce the notion of acceptable noncommutative random variables and investigate their essential properties. More precisely, we provide several efficient estimation of tail probabilities of sums of noncommutative random variables under some mild conditions. Moreover, we investigate the complete convergence of a sequence of the form [Formula: s...

In the quantum setting, there are several concepts of independence but they are insufficient for obtaining some noncommutative counterparts of maximal inequalities, and so we deal with the notion of noncommutative independence. We then employ a new approach to establish some maximal inequalities such as the strong and weak symmetrization, L\'evy, a...

In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties. We describe the interpolation with a parameter function of these spaces. As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund s...

We introduce near-martingales in the setting of quantum probability spaces and present a trick for investigating some of their properties. For instance, we give a near-martingale analogous result of the fact that the space of all bounded L p -martingales, equipped with the norm ∥ · ∥ p , is isometric to L p (M) for p > 1. We also present Doob and R...

It is known that the noncommutative Hardy spaces H1(M) and H1max(M) do not coincide, in general. In fact, it may happen that H1(M)⊈H1max(M). It is an interesting question whether the reverse inclusion holds or not. In this note, motivated by this question, we prove that the validity of inequality ‖x‖H1C(M)≤c‖x‖H1max(M) for some c and all martingale...

In this paper, we consider continuous parameter martingale Hardy–Lorentz spaces and describe their real interpolation spaces when we apply function parameter to Hardy–Lorentz and BMO spaces. Some new interpolation theorems concerning continuous parameter Hardy–Lorentz spaces are formulated. The results generalize some fundamental interpolation theo...

In this paper, we introduce the concept of $K$-fusion frames and propose the duality for such frames. The relation between the local frames of $K$-fusion frames with their dual is studied. The elements from the range of a bounded linear operator $K$ can be reconstructed by $K$-frames. Also, we establish $K$-fusion frame multipliers and investigate...

The main purpose of this paper is to identify the set of optimal dual frames under erasures. To this end, we characterize extreme points in the set of all optimal duals for 1-erasure. Also, we give some conditions under which an alternate dual frame is either not optimal or is a non-unique optimal dual. Moreover, we obtain a new characterization of...

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Haj\'ek--Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if $x_1, x_2, \ldots, x_n$ are successively independent self-adjoint random variables in a noncomm...

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Haj\'ek--Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if $x_1, x_2, \ldots, x_n$ are successively independent self-adjoint random variables in a noncomm...

In this paper, we introduce octadecic functional equation. Moreover, we prove the stability of the octadecic functional equation in multi-normed spaces by using the fixed point method.

We introduce martingale weighted twoparameter Lorentz spaces and establish atomic decomposition theorems. As an application of atomic decomposition we obtain a sufficient condition for sublinear operators defined on martingale weighted Lorentz spaces to be bounded. Moreover, some interpolation properties with a function parameter of those spaces ar...

We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

In this paper, we establish atomic decompositions for the martingale Hardy-Lorentz spaces. As an application, with the help of atomic decomposition, some interpolation theorems with a function parameter for these spaces are proved.

In this paper, we discuss the dual of a von Neumann–Schatten p-frames in separable Banach spaces and obtain some of their characterizations. Moreover, we present a classical perturbation result to von Neumann–Schatten p-frames.

In this paper, we present a fixed point method to prove generalized Hyers-Ulam stability of the systems of quadratic-cubic functional equations with constant coefficients in modular spaces.

We introduce and study the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that these spaces satisfy the uniform Opial condition with respect to p-a.e.-convergence for both the Luxemburg norm and the Amemiya norm....

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which inst...

In this paper, we present a fixed point method to prove generalized Hyers–Ulam stability of derivations in modular spaces.

We extend the classical Hoeffding inequality to the noncommutative setting. Under certain conditions we establish an improved version of the noncommutative Bennett inequality and use it to prove an extension of Rosenthal inequality as follows: Let $(\mathfrak{M}, \tau)$ be a noncommutative probability space, $\mathfrak{N}$ be a von Neumann subalgeb...

Let $\mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $\tau$. A closed densely defined operator $x$ affiliated with $\mathfrak{M}$ is called $\tau$-measurable if there exists a number $\lambda \geq 0$ such that $\tau \left(e^{|x|}(\lambda,\infty)\right)<\infty$. A number of useful...

In this paper, we present a fixed point method to prove generalized Hyers-Ulam stability of the generalized Jensen functional equation f (rx + sy) = rg(x) + sh(x) in modular spaces.

In this paper, we approximate the following additive functional inequality for all x11,..., xk d+1 Є X. We investigate homomorphisms in proper multi-CQ*-algebras and derivations on proper multi-CQ*-algebras associated with the above additive functional inequality.

In this paper, we investigate homomorphisms and derivations in proper JCQ∗JCQ∗-triples with the following functional equation: 1kf(kx+ky+kz)=f(x)+f(y)+f(z) for a fixed positive integer kk. We moreover prove the generalized Hyers–Ulam stability of homomorphisms in proper JCQ∗JCQ∗-triples and of derivations on proper JCQ∗JCQ∗-triples via a fixed poin...

We introduce the notion of modular G-metric spaces and obtain some fixed point theorems of contractive mappings defined on modular G-metric spaces.

We prove the existence and uniqueness of a common fixed point of compatible mappings of integral type in modular metric spaces. MSC: 47H09, 47H10, 46A80.

The purpose of this paper is to prove the existence of the unique common fixed point theorems of a pair of weakly compatible mappings satisfying $\Phi$-maps in modular G-metric spaces.

In this paper, we establish the Hyers-Ulam stability of the orthogonal quadratic functional equation of Pexiderized type f(x + y) + f(x - y) = 2g(x) + 2h(y), x ⊥ y in which ⊥ is the orthogonality in the sense of Rätz in modular spaces.

In this paper, the Hyers-Ulam stability of the Volterra integrodifferential equation x ' (t)=g(t,x(t))+∫ 0 t K(t,s,x(s))ds, and the Volterra equation x(t)=g(t,x(t))+∫ 0 t K(t,s,x(s))ds, on the finite interval [0,T], T>0, are studied, where the state x(t) takes values in a Banach space X.

In this paper, we introduce the notion of a frame in a 2- inner product space and give some characterizations. These frames can be considered as a usual frame in a Hilbert space, so they share many useful properties with frames. © 2013 Academic Center for Education, Culture and Research TMU.

In this paper, we consider non-commutative Orlicz spaces as modular spaces and show that they are complete with respect to their modular. We prove some convergence theorems for ττ-measurable operators and deal with uniform convexity of non-commutative Orlicz spaces.

In this paper we introduce the notion of a von Neumann-Schatten p-frame in separable Banach spaces and obtain some of their characterizations. We show that p-frames and g-frames are a class of von Neumann-Schatten p-frames.

As for an orthonormal basis, a frame allows each element in the underlying Hilbert space to be written as an unconditionally convergent infinite linear combination of the frame elements. The coefficients are called frame coefficients. Peter G. Casazza and Ole Chris-tensen introduced some methods to approximate frame coefficients. In this article, w...

At first we find the solution of the functional equation D f ( x 1 , … , x m ) : = ∑ k = 2 m ( ∑ i 1 = 2 k ∑ i 2 = i 1 + 1 k + 1 ⋯ ∑ i m - k + 1 = i m - k + 1 m ) f ( ∑ i = 1 , i i 1 , … , i m - k + 1 m x i - ∑ r = 1 m - k + 1 x i r ) + f ( ∑ i = 1 m x i ) - 2 m - 1 f ( x 1 ) = 0, where m ≥ 2 is an integer number. Then, we obtain the generalized Hy...

We prove the existence of fixed point and uniqueness of quasi-contractive mappings in modular metric spaces which was introduced by Ćirić

In this article, we prove the nonlinear stability of the quartic functional equation 1 6 f ( x + 4 y ) + f ( 4 x - y ) = 3 0 6 9 f x + y 3 + f ( x + 2 y ) (1) + 1 3 6 f ( x - y ) - 1 3 9 4 f ( x + y ) + 4 2 5 f ( y ) - 1 5 3 0 f ( x ) (2) (3) in the setting of random normed spaces Furthermore, the interdisciplinary relation among the theory of rand...

In this paper, we prove the stability of Euler-Lagrange quadratic map-pings in the framework of non-Archimedean normed spaces. Our results in the setting of non-Archimedean normed spaces are different from the results in the setting of normed spaces.

The generalized stability of the Euler-Lagrange quadratic mappings in the framework of non-Archimedean random normed spaces is proved. The interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, and the theory of functional equations is presented. Key wordsgeneralized Hyers-Ulam stability–Euler-Lagrange...

In this paper, using the fixed point method, we prove the stability of a generalized functional equation in the Forti spirit.

Let X, Y be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping f: X → Y satisfies f(x + iy) + f(x - iy) = 2f(x) - 2f(y) for all x, y ∈ X, then the mapping f: X → Y satisfies f(x + y) + f(x - y) = 2f(x) + 2f(y) for all x, y ∈ X. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation (1)...

In this paper, we present a Mazur–Ulam type theorem in non-Archimedean strictly convex 2-normed spaces and present some properties of mappings on non-Archimedean strictly 2-convex 2-normed spaces.

In this study, the stability of the cubic functional equation: f(2x+y)+f(2x-y) = 2f(x+y)+2f(x-y)+ 12f(x) in the setting of Menger probabilistic normed spaces is proved.

Let T, A be operators with domains D(T) subset of D(A) in a normed space X. The operator A is called T-bounded if parallel to Ax parallel to <= a parallel to x parallel to + b parallel to Tx parallel to for some a, b >= 0 and all x is an element of D(T). If A has the Hyers-Ulam stability then under some suitable assumptions we show that both T and...

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.

A quasi norm is a non-negative function parallel to.parallel to on a linear space (sic) satisfying the same axioms as a norm except for the triangle inequality, which is replaced by the weaker condition that "there is a constant K >= I such that parallel to x + y parallel to <= K(parallel to x parallel to + parallel to y parallel to) for all x. y i...

Let U be a C*-algebra and B be a von Neumann algebra that both act on a Hilbert space H. Let M and N be inner product modules over U and B, respectively. Under certain assumptions, we show that for each mapping f : M -> N satisfying parallel to vertical bar < f(x), f(y)>vertical bar-vertical bar < x,y >vertical bar parallel to <= phi(x, y) (x, y is...

Let $\A$ be a $C^*$-algebra and $\B$ be a von Neumann algebra that both act on a Hilbert space $\Ha$. Let $\M$ and $\N$ be inner product modules over $\A$ and $\B$, respectively. Under certain assumptions we show that for each mapping $f\colon{\mathcal M} \to {\mathcal N}$ satisfying $$\||\ip{f(x)}{f(y)}|-|\ip{x}{y}| \|\leq\phi(x,y)\qquad (x,y\in{\...

In this paper we prove the generalized Hyers–Ulam–Rassias stability of extended derivations on unital Banach algebras associated to a generalized Jensen equation.

We prove the generalized stability of the cubic type functional equation $$f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)$$ and another functional equation $$f(ax+y)+f(x+ay)=(a+1)(a-1)^{2}[f(x)+f(y)] +a(a+1)f(x+y),$$ where $a$ is an integer with $a \neq 0, \pm 1$ in the framework of non-Archimedean normed spaces.