The main goal of this paper is to present new bounds for the norms and numerical radii of certain Hilbert space operators, which involve the sums of operators' products. The …
The main goal of this paper is to present new bounds for the norms and numerical radii of certain Hilbert space operators, which involve the sums of operators' products. The obtained result will be utilized to obtain some equivalent conditions regarding certain operator identities, to find upper bounds for the sum of two operators, to obtain certain refinements of celebrated numerical radius bounds, and to present a new reverse for the triangle inequality when positive operators are treated.
Abstract In this paper, we prove some bounds for the ‐numerical radius as natural extensions of certain known bounds for the numerical radius. This complements many recent studies in this …
Abstract In this paper, we prove some bounds for the ‐numerical radius as natural extensions of certain known bounds for the numerical radius. This complements many recent studies in this direction. Among many results, we show that if is an complex matrix, with Aluthge transform , then for , where and denote the ‐numerical radius and the Schatten ‐norm, respectively. This extends a celebrated result by Yamazaki, who showed the same result when .
In this paper, we present several new bounds in normed spaces, with applications towards subadditivity behavior and angular distances, in these spaces, with an additional application towards Hilbert space operators. …
In this paper, we present several new bounds in normed spaces, with applications towards subadditivity behavior and angular distances, in these spaces, with an additional application towards Hilbert space operators. The main tool we use to establish our results is the treatment of convex functions and their properties.
Finding new bounds for norms and the numerical radius of certain matrix forms is a renowned topic that has attracted numerous researchers. In this paper, we show some new bounds …
Finding new bounds for norms and the numerical radius of certain matrix forms is a renowned topic that has attracted numerous researchers. In this paper, we show some new bounds for unitarily invariant norms and the numerical radius of certain matrix forms that involve products of matrices and sums of products. New arithmetic-geometric mean-type inequalities and a refinement of the celebrated Cauchy–Schwarz inequality are shown among the obtained results. Our results are compared with those in the literature through numerical examples and rigorous approaches.
To better understand the algebra \(\mathcal{M}_n\) of all \(n\times n\) complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to …
To better understand the algebra \(\mathcal{M}_n\) of all \(n\times n\) complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. Among many results, we present order-preserving results, Choi–Davis-type inequalities, mean-convex inequalities, sub-multiplicative results for the real part, and new bounds of the absolute value of accretive matrices. These results will be compared with the existing literature. In the end, we quickly pass through related entropy results for accretive matrices.
This paper's main objective is to find new upper bounds for the norm of the sum of two Hilbert space operators and their Kronecker product. The obtained results extend some …
This paper's main objective is to find new upper bounds for the norm of the sum of two Hilbert space operators and their Kronecker product. The obtained results extend some previously known results from the set of positive operators to arbitrary ones and refine several existing bounds. In particular, applications of the established bounds include refining celebrated numerical radius inequalities and the celebrated operator Cauchy–Schwarz norm inequality.
In this article, we discuss some problems related to adjointable operators between two Hilbert modules over a $C^*$-algebra $\mathfrak{A}$. In particular, we discuss the positivity of block matrices in this …
In this article, we discuss some problems related to adjointable operators between two Hilbert modules over a $C^*$-algebra $\mathfrak{A}$. In particular, we discuss the positivity of block matrices in this class, where equivalent conditions are provided. Then, we generalize the mixed Schwarz inequality for the operators mentioned above. This generalization's significance arises because the mixed Schwarz inequality for bounded linear operators on Hilbert spaces does not apply for adjointable operators between Hilbert $\mathfrak{A}$-modules. Then, some applications that involve the numerical radius of Hilbert space operators will be provided, where the positivity of certain block matrices is essential. Numerical examples to support the validity and advantage of the obtained results are given in the end.
In this paper, we show some inner product inequalities, which can be considered of Cauchy-Schwarz and Buzano type inequalities for Hilbert space operators. This will lead to several applications that …
In this paper, we show some inner product inequalities, which can be considered of Cauchy-Schwarz and Buzano type inequalities for Hilbert space operators. This will lead to several applications that include a new equivalent statement to the fact that an operator matrix is positive and new upper bounds for the numerical radius.
Fejér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. …
Fejér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For the matrix Levin-Stečkin type, we present more rigorous results involving the partial Löewner ordering for Hermitian matrices. Further related results involving synchronous functions are presented, too.
The main goal of this paper is to present new inequalities for convex and log-convex functions. The significance of these inequalities follows from the way they extend many known results …
The main goal of this paper is to present new inequalities for convex and log-convex functions. The significance of these inequalities follows from the way they extend many known results in the literature concerning convex functions, log-convex functions, means comparisons and matrix inequalities.
Scalar quantities associated with Hilbert-space operators have attracted the attention of numerous researchers due to their role in understanding the geometry of the C * -algebra of bounded linear operators …
Scalar quantities associated with Hilbert-space operators have attracted the attention of numerous researchers due to their role in understanding the geometry of the C * -algebra of bounded linear operators on a Hilbert space.In this paper, we explore the Berezin number of operator matrices, and present several new relations that simulate the existing relations between the numerical radius and the operator norm.
The main goal of this article is to present multiple term refinements of the well-known Jensen's inequality for h-convex functions for a non-negative super-multiplicative and super-additive function h.For example, we …
The main goal of this article is to present multiple term refinements of the well-known Jensen's inequality for h-convex functions for a non-negative super-multiplicative and super-additive function h.For example, we show thatfor the h-convex function f and certain positive summands.The significance of the obtained results is the way they extend known results from the setting of convex functions to other classes of functions.
The main focus of this paper is a study of Jensen-type inequalities for the Lipschitzian functions.We establish the reverse of the Jensen inequality expressed in terms of the corresponding Lipschitz …
The main focus of this paper is a study of Jensen-type inequalities for the Lipschitzian functions.We establish the reverse of the Jensen inequality expressed in terms of the corresponding Lipschitz constant.In addition, we also obtain the reverse of the superadditivity relation for a convex function, expressed in the same way.As an application, we obtain reverses of power mean inequalities, the Hölder inequality, and the Hermite-Hadamard inequality, expressed in terms of the Lipschitzianity.In particular, we derive reverses of the arithmetic-geometric mean inequality in both difference and quotient forms.
This paper investigates possible upper and lower bounds for the numerical radii of 2 × 2 operator matrices.The obtained results improve and generalize many results from the literature in accessible …
This paper investigates possible upper and lower bounds for the numerical radii of 2 × 2 operator matrices.The obtained results improve and generalize many results from the literature in accessible forms.Moreover, many numerical examples will be given to support the feasibility of our findings.
This paper finds new upper bounds for the singular values of certain operator forms.Compared with the existing literature, numerous numerical examples will be given to show that the obtained forms …
This paper finds new upper bounds for the singular values of certain operator forms.Compared with the existing literature, numerous numerical examples will be given to show that the obtained forms add a new set of independent bounds, that are incomparable with some celebrated known results.
Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms …
Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.
Convex functions and their analogues have been powerful tools in almost all mathematical fields, including optimization, fractional calculus, mathematical analysis, functional analysis, operator theory, and mathematical physics. It is well …
Convex functions and their analogues have been powerful tools in almost all mathematical fields, including optimization, fractional calculus, mathematical analysis, functional analysis, operator theory, and mathematical physics. It is well established in the literature that a convex function $f:[0,\infty)\to [0,\infty)$ with $f(0)=0$ is necessarily superadditive, while a concave function $f:[0,\infty)\to [0,\infty)$ is subadditive. The converses of these two assertions are not valid in general. The main target of this article is to study the subadditivity and superadditivity of convex and superquadratic functions. In particular, we obtain several results extending, refining, and reversing some known inequalities in this direction. Further discussion of superquadratic functions in this line will be given.
AbstractExtending certain scalar and norm inequalities, we present new inequalities for the numerical radius, which generalize and refine some known results. Applications of the obtained inequalities include a new original …
AbstractExtending certain scalar and norm inequalities, we present new inequalities for the numerical radius, which generalize and refine some known results. Applications of the obtained inequalities include a new original proof of the matrix arithmetic-geometric mean inequality and certain extensions of some well-established results from the literature for products of matrices.KEYWORDS: Convex functionnorm inequalitynumerical radiusMATHEMATICS SUBJECT CLASSIFICATION: Primary: 15A60Secondary: 47A1247A30 Authors' contributionsThe authors have contributed equally to this work.Disclosure statementAll authors declare that they have no conflicts of interest.Additional informationFundingThe authors did not receive any funding to accomplish this work.
AbstractThe class of geometrically convex functions is a rich class that contains some important functions. In this paper, we further explore this class and present many interesting new properties, including …
AbstractThe class of geometrically convex functions is a rich class that contains some important functions. In this paper, we further explore this class and present many interesting new properties, including fundamental inequalities, supermultiplicative type inequalities, Jensen-Mercer inequality, integral inequalities, and refined forms. The obtained results extend some celebrated results from the context of convexity to geometric convexity, with interesting applications to numerical inequalities for the hyperbolic and exponential functions.Mathematics Subject Classification (2020): 26A5126D1526E60Key words: Geometrically-convex functionHermite-Hadamard inequalitydoubly-convex functions
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 …
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.
The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. …
The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented.
Abstract In this paper, we present several new bounds in normed spaces, with applications toward subadditivity behavior and angular distances, in these spaces, with an additional application towards Hilbert space …
Abstract In this paper, we present several new bounds in normed spaces, with applications toward subadditivity behavior and angular distances, in these spaces, with an additional application towards Hilbert space operators. The main tool we use to establish our results is the treatment of convex functions and their properties.
Abstract This paper shows a new mixed Cauchy-Schwarz inequality for Hilbert-space operators. In particular, we show that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$, and …
Abstract This paper shows a new mixed Cauchy-Schwarz inequality for Hilbert-space operators. In particular, we show that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$, and if $x,y\in\mathcal{H}$, then \[\left| \left\langle Tx,y \right\rangle \right|\le \sqrt{\left\langle {{\left| {{T}^{*}} \right|}^{\frac{3-t}{2}}}x,x \right\rangle \left\langle {{\left| T \right|}^{\frac{1+t}{2}}}y,y \right\rangle };\text{ }\left( 0\le t\le 1 \right),\] where $\left| T \right|={{\left( {{T}^{*}}T \right)}^{\frac{1}{2}}}$, and $T^*$ is the adjoint operator of $T$. We also present several applications of the obtained results, where new bounds for the numerical radius will be shown.
Abstract In this paper, we develop a concept of a logarithmically super\-qua\-dratic function. Such a class of functions is defined via superquadratic functions.These functions possess some superior properties, especially if …
Abstract In this paper, we develop a concept of a logarithmically super\-qua\-dratic function. Such a class of functions is defined via superquadratic functions.These functions possess some superior properties, especially if they take values greater than or equal to one. We prove that they are convex and superadditive in the latter case.In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Furthermore, we derive an external form of the Jenseninequality and the corresponding reverse. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions.All established results are derived via the corresponding relations for superquadratic functions.
It is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ {1 \over 4}\| {A^* A + AA^* } \| \le ( {w(A …
It is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ {1 \over 4}\| {A^* A + AA^* } \| \le ( {w(A )} )^2 \le {1 \over 2}\| {A^* A + AA^* }\| , $$ where $w(\cdot )$ and $\| \cdot \| $ are the numerical radius and the u
It is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ w(A) \le \frac{1}{2} (\| A \| + \| A^2 \|^{1/2} ), $$ …
It is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ w(A) \le \frac{1}{2} (\| A \| + \| A^2 \|^{1/2} ), $$ where $w(A)$ and $\|A\|$ are the numerical radius and the usual operator norm of $A$, respectively. An applicati
We give an inequality relating the operator norm of $T$ and the numerical radii of $T$ and its Aluthge transform. It is a more precise estimate of the numerical radius …
We give an inequality relating the operator norm of $T$ and the numerical radii of $T$ and its Aluthge transform. It is a more precise estimate of the numerical radius than Kittaneh's result [Studia Math. 158 (2003)]. Then we obtain an equivalent conditio
We give several sharp inequalities involving powers of the numerical radii and the usual operator norms of Hilbert space operators. These inequalities, which are based on some classical convexity inequalities …
We give several sharp inequalities involving powers of the numerical radii and the usual operator norms of Hilbert space operators. These inequalities, which are based on some classical convexity inequalities for nonnegative real numbers and some operator
In this article, we introduce the concept of harmonically log-convex functions, which seems to be strongly connected to unitarily invariant norms.Then, we prove Hermite-Hadamard inequalities for these functions.As an application, …
In this article, we introduce the concept of harmonically log-convex functions, which seems to be strongly connected to unitarily invariant norms.Then, we prove Hermite-Hadamard inequalities for these functions.As an application, we present many inequalities for the trace operator and unitarily invariant norms.
Several inequalities for Hilbert space operators are extended. These include results of Furuta, Halmos, and Kato on the mixed Schwarz inequality, the generalized Reid inequality as proved by Halmos and …
Several inequalities for Hilbert space operators are extended. These include results of Furuta, Halmos, and Kato on the mixed Schwarz inequality, the generalized Reid inequality as proved by Halmos and a classical inequality in the theory of compact non-self-adjoint operators which is essentially due to Weyl. Some related inequalities are also discussed.
In a recent work of the authors, we showed some general inequalities governing numerical radius inequalities using convex functions. In this article, we present results that complement the aforementioned inequalities. …
In a recent work of the authors, we showed some general inequalities governing numerical radius inequalities using convex functions. In this article, we present results that complement the aforementioned inequalities. In particular, the new versions can be looked at as refined and generalized forms of some well-known numerical radius inequalities. Among many other results, we show that fA∗A+AA∗4≤∫01f1−tB2+tC2dt≤fw2A, where A is a bounded linear operator on a Hilbert space having the Cartesian decomposition A = B + iC and f:[0,∞)→[0,∞) is an increasing operator convex function. This result, for example, extends and refines a celebrated result by Kittaneh.
The main objective of the present paper is to estimate the precision of Jensen-Steffensen inequality. We obtain results that complement, generalize, unify and agree with some of the previously known …
The main objective of the present paper is to estimate the precision of Jensen-Steffensen inequality. We obtain results that complement, generalize, unify and agree with some of the previously known results in this area.
In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine …
In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if A is a bounded linear operator on a complex Hilbert space, then w2(A)≤∫01tA+1−tA∗2dt≤12A2+A∗2 where w(A) and A are the numerical radius and the usual operator norm of A, respectively.
In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and …
In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds. Among many other applications, it is shown that if T is accretive-dissipative, then 1/?2 ||T|| ? ?(T), where ?(?) and ||?||denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality 1/2 ||T|| ? ?(T).
New upper and lower bounds for the numerical radii of Hilbert space operators are given.An application to involution operators is also provided.
New upper and lower bounds for the numerical radii of Hilbert space operators are given.An application to involution operators is also provided.
AbstractWe study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such …
AbstractWe study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such matrices and we establish an operator norm bound in this context.Keywords: numerical rangegeometric meanprincipal powersymbolic calculusoperator normAMS Subject Classification: 15A45
In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's …
In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality, the arithmetic-harmonic and the geometric-harmonic mean inequalities.
New inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen's inequality are established. Particular instances in the case of normed spaces and …
New inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen's inequality are established. Particular instances in the case of normed spaces and for complex and real n -tuples are given. Refinements of Shannon's inequality and the positivity of Kullback-Leibler divergence are obtained.
In this article we present refinements of Jensen's inequality and its reversal for convex functions, by adding as many refining terms as we wish.Then as a standard application, we present …
In this article we present refinements of Jensen's inequality and its reversal for convex functions, by adding as many refining terms as we wish.Then as a standard application, we present several refinements and reverses of well known mean inequalities.
The Specht ratio S(h) is the optimal constant in the reverse of the arithmetic-geometric mean inequality, i.e., if 0 < m a,b M anda and r = min{μ,1 -μ} .In …
The Specht ratio S(h) is the optimal constant in the reverse of the arithmetic-geometric mean inequality, i.e., if 0 < m a,b M anda and r = min{μ,1 -μ} .In this paper, we improve it by virtue of the Kantorovich constant, utilizing the refined scalar Young inequality we establish a weighted arithmetic-geometric-harmonic mean inequality for two positive operators.In the remainder of this work we focus on extending the refined weighted arithmetic-harmonic mean inequality to an operator version for another type of improvement.
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers …
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.
This note aims to present some scalar inequalities and operator inequalities on a Hilbert space. Firstly, the direct reverse weighted arithmetic-harmonic mean inequalities for scalars are obtained. Secondly, based on …
This note aims to present some scalar inequalities and operator inequalities on a Hilbert space. Firstly, the direct reverse weighted arithmetic-harmonic mean inequalities for scalars are obtained. Secondly, based on these scalar inequalities, the corresponding operator inequalities are established. Finally, we present the mixed arithmetic-geometric and geometric-harmonic means inequalities for two positive operators.
We give reverses of the classical Young inequality for positive real numbers and we use these to establish reverse Young and Heinz inequalities for matrices.
We give reverses of the classical Young inequality for positive real numbers and we use these to establish reverse Young and Heinz inequalities for matrices.
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. …
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.
The main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily …
The main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily invariant norms. In particular, variants, but weaker forms, of the well-known Heinz inequality and its generalizations are extended to the context of numerical radius.
For any Hilbert space 3C, let 3C be the totality of bounded selfadjoint operators with spectrum contained in an interval I, which need not be finite. If f is a …
For any Hilbert space 3C, let 3C be the totality of bounded selfadjoint operators with spectrum contained in an interval I, which need not be finite. If f is a function from 3C to the self-adjoint operators on SC obtained from a bounded real-valued function fo on I by the spectral theorem (that is, by f(A) = fJ ,fo9(X))dEx, where Ex is the spectral resolution of A), thenf is called an operator function (associated with I). An operator functionf is defined, oncefo is given, for all such 3C. An operator function f associated with I is convex provided A, B in 3C and O t?< 1 imply