arXiv subjects

Mohammad Sababheh

Publications and source records attributed to Mohammad Sababheh.

At least 19 recordsLinked to original sources

arXiv2026

Some Spectral Problems for First Order Normal Differential Operators in the Weighted Hilbert Spaces of Vector-Functions

In this article, in order to the minimal operator generated by the first-order differential-operator expression in the weighted Hilbert space of vector functions in the finite interval to be formal normal, the relationship between the variable operator coefficient of this differential-operator expression and the weight function is established. Afterwards, the general form of all normal extension of the minimal operator is found using the Glazman-Krein-Naimark Method. Then, the structure of spectrum of such extensions is investigated. Later on, the issue of belonging to Schatten von-Neumann classes is explored, as well as the asymptotic behaviour of the singular numbers of the inverse of such normal extensions. Lastly, an approach is developed on all normal extension expressed in the weighted Hilbert spaces.

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arXiv2026

Integral Numerical Radius and Operator Matrix Bounds

We establish new integral inequalities for the numerical radius and the operator norm of bounded linear operators on Hilbert spaces. Our results refine classical triangle-type and operator matrix inequalities by incorporating convex combinations and integral averaging techniques. Several consequences, including new identities, sharper bounds, and equality conditions, are obtained, revealing deeper structural connections between the numerical radius and operator norm.

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arXiv2026

A new collection of $n-$tuple operator inequalities

In this paper, we present several new bounds for the norm and numerical radius of sums of Hilbert space operators. The obtained bounds form a new collection that enriches our understanding of these bounds. We compare our bounds with the existing literature using examples that demonstrate, in general, how our results are incomparable with the known bounds. Of particular interest are the treatment of the triangle inequality, the numerical radius of operator matrices, and singular value bounds for sums of operators.

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arXiv2026

More inner product bounds with applications

The main goal of this paper is to present new bounds for certain inner products in Hilbert spaces, with applications to the numerical radius and the operator norm. The obtained results significantly improve earlier results in this direction.

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arXiv2025

The numerical radius of fractional powers of matrices

Using integral representations of the fractional power of matrices, and the geometric intuition of sectorial matrices, we show that for any accretive-dissipative matrix $A$ and any $t \in (0,1)$, the matrix $A^t$ is accretive-dissipative, and that \[ ω(A^t)\geq ω^t(A) , \] where $ω(\cdot)$ is the numerical radius. This inequality complements the well-known power inequality $ω(A^k)\leq ω^k(A)$, valid for any square matrix and positive integer power $k$. As an application, we prove that if $A$ is accretive, then the above fractional inequality holds if $0<t<\frac{1}{2}$. Other consequences will be given too.

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arXiv2025

The Resolvent Mean and The Parametrized $\mathcal{A} \sharp \mathcal{B}$

Resolvent average and weighted $\mathcal{A}\sharp \mathcal{H}$-mean have been defined recently for positive definite matrices. Since the class of accretive matrices provides a general framework for addressing certain known results on positive matrices, this paper extends the notions of resolvent average and the weighted $\mathcal{A}\sharp \mathcal{H}$-mean to accretive matrices and discusses some of their properties.\\ The obtained results happen to be legitimate generalizations of those known results on positive definite matrices.\\ Among many results, we show that if $A,B$ are positive definite matrices, and $0\leqλ\leq 1, μ>0$, then \[\mathcal{R}_μ(A,B,1-λ,λ)+μI \geq C \Big(A \sharp_{\bmλ} B +μI\Big),\] where $\mathcal{R}_λ$ is the resolvent average, $\sharp_{\bmλ}$ is the weighted geometric mean and $I$ is the identity matrix, for some positive constant $C$; as a new relation between the resolvent average and the geometric mean.

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arXiv2025

Accretive Partial Transpose Matrices and Their Connections to Matrix Means

Accretive partial transpose (APT) matrices have been recently defined, as a natural extension of positive partial transpose (PPT) matrices. In this paper, we discuss further properties of APT matrices in a way that extends some of those properties known for PPT matrices. Among many results, we show that if $A,B,X$ are $n\times n$ complex matrices such that $A,B$ are sectorial with sector angle $α$ for some $α\in [0,π/2)$, and if $f:(0,\infty)\to(0,\infty)$ is a certain operator monotone function such that $\begin{bmatrix} \cos^2(α) f(A) & X X^* & \cos^2(α) f(B) \end{bmatrix}$ is APT, Then $\begin{bmatrix} f(A)\nabla_t f(B) & X X^* & f(A \nabla_tB ) \end{bmatrix}$ is APT for any $0\leq t\leq 1$, where $\nabla_t$ is the weighted arithmetic mean.

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arXiv2024

Some inequalities for adjointable operators on Hilbert $C^*$-modules

The main purpose of this paper is, in the general setting of the adjointable operators on Hilbert $C^*$-modules, to develop two new tools that can be applied to deal with the positive solutions of certain operator equations, the operator norm as well as the numerical radius, respectively. Among other things, the positivity of a $2\times 2$ block operator matrix is clarified without any preconditions on its entries, and a generalized version of the mixed Schwarz inequality with a parameter is derived. Numerical examples are provided to illustrate the non-triviality of this newly obtained inequality.

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arXiv2024

On a binary operation for positive operators

M. Lin defined a binary operation for two positive semi-definite matrices in studying certain determinantal inequalities that arise from diffusion tensor imaging. This operation enjoys some interesting properties similar to the operator geometric mean. We study this operation further and present numerous properties emphasizing the relationship with the operator geometric mean. In the end, we present an application toward Tsallis relative operator entropy.

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arXiv2023

Numerical Radius Bounds via the Euclidean Operator Radius and Norm

In this paper, we begin by showing a new generalization of the celebrated Cauchy-Schwarz inequality for the inner product. Then, this generalization is used to present some bounds for the Euclidean operator radius and the Euclidean operator norm. These bounds will be used then to obtain some bounds for the numerical radius in a way that extends many well-known results in many cases. The obtained results will be compared with the existing literature through numerical examples and rigorous approaches, whoever is applicable. In this context, more than 15 numerical examples will be given to support the advantage of our findings. Among many consequences, will show that if $T$ is an accretive-dissipative bounded linear operator on a Hilbert space, then ${{\left\| \left( \Re T,\Im T \right) \right\|}_{e}}=ω\left( T \right)$, where $ω(\cdot), \|(\cdot,\cdot)\|_e, \Re T$ and $\Im T$ denote, respectively, the numerical radius, the Euclidean norm, the real part and the imaginary part.

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arXiv2023

On the Euclidean operator radius and norm

In this paper, we show several bounds for the numerical radius of a Hilbert space operator in terms of the Euclidean operator norm. The obtained forms will enable us to find interesting refinements of celebrated results in the literature. Then, the $f$-operator radius, recently defined as a generalization of the Euclidean operator radius, will be studied. Many upper bounds will be found and matched with existing results that treat the numerical radius. Special cases of this discussion will lead to some refinements and generalizations of some well-established results in the field. Further, numerical examples are given to support our findings, and a simple optimization application will be presented.

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arXiv2023

Operator Spectral Geometric Versus Geometric Mean

The main goal of this article is to present new inequalities for the spectral geometric mean $A\natural_t B$ of two positive definite operators $A, B$ on a Hilbert space. The obtained results complement many known inequalities for the geometric mean $A\sharp_t B$. In particular, explicit comparisons between $A\natural_t B$ and $A\sharp_t B$ are given, with applications towards Ando-type inequalities and Ando-Hiai inequalities for $A\natural_t B$ and some other consequences.

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arXiv2023

New Orders Among Hilbert Space Operators

This article introduces several new relations among related Hilbert space operators. In particular, we prove some Löewner partial orderings among $T, |T|, \mathcal{R}T, \mathcal{I}T, |T|+|T^*|$ and many other related forms, as a new discussion in this field; where $\mathcal{R}T$ and $\mathcal{I}T$ are the real and imaginary parts of the operator $T$. Our approach will be based on proving the positivity of some new matrix operators, where several new forms for positive matrix operators will be presented as a key tool in obtaining the other ordering results. As an application, we present some results treating numerical radius inequalities in a way that extends some known results in this direction, in addition to some results about the singular values.

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arXiv2023

On the matrix Cauchy-Schwarz inequality

The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that \[\left\| T \right\|\le \frac{1}{4}\left( \left\| \left| T \right|+\left| {{T}^{*}} \right|+2\mathfrak RT \right\|+\left\| \left| T \right|+\left| {{T}^{*}} \right|-2\mathfrak RT \right\| \right),\] where $\mathfrak RT$ is the real part of $T$.

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arXiv2023

A convex-block approach for numerical radius inequalities

This article implements a simple convex approach and block techniques to obtain several new refined versions of numerical radius inequalities for Hilbert space operators. This includes comparisons among the norms of the operators, their Cartesian parts, their numerical radii, the numerical radius of the product of two operators, and the Aluthge transform.

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arXiv2023

Operator inequalities via accretive transforms

In this article, we employ certain properties of the transform $C_{M,m}(A)=(MI-A^*)(A-mI)$ to obtain new inequalities for the bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$. In particular, we obtain new relations among $|A|,|A^*|,|\mathfrak{R}A|$ and $|\mathfrak{I}A|$. Further numerical radius inequalities that extend some known inequalities will be presented too.

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arXiv2022

Further Properties of PPT and Hyponormal Matrices

This paper discusses further properties of positive partial transpose matrices, with applications towards hyponormal, semi-hyponormal, and $(α,β)$-normal matrices. The obtained results present extensions and improvements of many results in the literature.

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arXiv2022

Generalized Euclidean Operator Radius

In this paper, we introduce the $f-$operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the $q-$operator radius. Properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

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