# operator norm and numerical radius

## Top PDF operator norm and numerical radius: ### Numerical Radius and Operator Norm Inequalities

In Section 2 of this paper, we establish a general numerical radius inequality that generalizes 1.6, 1.7, 1.8, and 1.9, from which numerical radius inequalities for sums, products, and commutators of operators are obtained. Usual operator norm inequalities that generalize 1.11 and related to 1.13 are presented in Section 3. ### Norm and Numerical Radius Inequalities for Sums of Bounded Linear Operators in Hilbert Spaces

The aim of this paper is to establish various new inequalities for the operator norm and numerical radius of sums of bounded linear operators in Hilbert spaces. In particular, two refinements of the generalised triangle inequality for operator norm are obtained. Particular cases of interest for two bounded linear operators and their applications for the Cartesian decomposition of an operator are also considered. ### Semi-Inner Products and the Numerical Radius of Bounded Linear Operators in Hilbert Spaces

Abstract. The main aim of this paper is to establish some connections that exist between the numerical radius w (A) , operator norm kAk and the semi- inner products hA, Ii p,n and hA, Ii p,w with p ∈ {i, s} that can be naturally defined on the Banach algebra B (H ) of all bounded linear operators defined on a Hilbert space H. Reverse inequalities that provide upper bounds for the nonnegative quantities kAk − w (A) and w (A) − hA, Ii p,n under various as- sumptions for the operator A are also given. ### New norm equalities and inequalities for operator matrices

Let LB(H) denote the C ∗ -algebra of all linear bounded operators on a complex separable Hilbert space H with inner product ·, ·. For M ∈ LB(H), let ω(M) = sup {|Mx, x| : x ∈ H and x =  } and M = sup {| Mx, y | : x, y ∈ H and x = y =  } denote the numerical radius and the usual operator norm of M, respectively. It is well known that ω( · ) deﬁnes a norm on LB(H), which is equivalent to the usual operator norm · . In fact, for every M ∈ LB(H), ### Inequalities for the Norm and the Numerical Radius of Linear Operators in Hilbert Spaces

The main aim of this paper is to establish other inequalities between the operator norm and its numerical radius. We employ, amongst others, the Buzano inequality as well as some results for vectors in inner product spaces due to Goldstein-Ryff- Clarke , Dragomir-S´ andor  and Dragomir . ### Some vector inequalities for two operators in Hilbert spaces with applications

Motivated by the above results we establish in this paper some vector in- equalities for two operators A, B for which the operator Re (B ∗ A) is nonnega- tive in the operator order that are related to the inequality (6). Applications for norm and numerical radius inequalities are provided as well. ### Upper Bounds for the Euclidean Operator Radius and Applications

Abstract. The main aim of the present paper is to establish various sharp upper bounds for the Euclidean operator radius of an n−tuple of bounded linear operators on a Hilbert space. The tools used are provided by several generalisations of Bessel inequality due to Boas-Bellman, Bombieri and the author. Natural applications for the norm and the numerical radius of bounded linear operators on Hilbert spaces are also given. ### A Note on Numerical Radius Operator Spaces

Recently, some new algebraic structures derived from operator spaces also have been intensively studied. An operator system is a matrix ordered operator space which plays a profound role in mathematical physics. Kavruk, Paulsen, Todorov and Tomforde gave a systematic study of tensor products and local property of operator systems in  . In , Luthra and Kumar showed that an operator system is exact if and only if it can be embedded into a Cuntz algebra. The numerical radius operator space is also an important algebraic structure which is introduced by Itoh and Nagisa  . The conditions to be a numerical radius space are weaker than the Ruan’s axiom for an operator space. It is shown that there is a  -complete isometry from a numerical radius oper- ator space into a Hilbert space with numerical radius norm. They also studied many relations between the operator spaces and the numerical radius operator spaces. The category of operator space can be regarded as a subcategory of nu- merical radius operator space. ### Some Inequalities of the Grüss Type for the Numerical Radius of Bounded Linear Operators in Hilbert Spaces

Motivated by the natural question that arise in order to compare the quantity w (AB) with other expressions comprising the norm or the numerical radius of the involved operators A and B (or certain expressions constructed with these operators), we establish in this paper some natural inequalities of the form ### Operator valued Extensions of Matrix norm Inequalities

Abstract. We describe a rather striking extension of a wide class of inequalities. Some famous classical inequalities, such as those of Hardy and Hilbert, equate to the evaluation of the norm of a matrix operator. Such inequalities can be presented in two versions, linear and bilinear. We show that in all such inequalities, the scalars can be replaced by operators on a Hilbert space, with the conclusions taking the form of an operator inequality in the usual sense. With careful formulation, a similar extension applies to the Cauchy–Schwarz inequality. ### BMO and Lipschitz norm estimates for the composition of Green’s operator and the potential operator

A-harmonic equation (.) in a bounded convex domain , G be Green’s operator and P be the potential operator deﬁned in equation (.) with the kernel K (x, y) satisfying the condition () of the standard estimates (.). Then there exists a constant C, independent of , such that ### Estimates of the modular type operator norm of the general geometric mean operator

In this paper, the modular-type operator norm of the general geometric mean operator over spherical cones is investigated. We give two applications of a new limit process, introduced by the present authors, to the establishment of Pólya-Knopp-type inequalities. We not only partially generalize the suﬃcient parts of Persson-Stepanov’s and Wedestig’s results, but we also provide new proofs to these results. ### Sesquilinear version of numerical range and numerical radius

During the past decades, several definitions of the numerical range in various settings have been introduced by many mathematicians. For instance, Marcus and Wang  opened the concept of the rth permanent numerical range of operator A. Furthermore, Descloux in  defined the notion of the essential numerical range of an operator with respect to a coercive sesquilinear form. In 1977, Marvin  and in 1984, independently, Tsing  introduce and characterize a new version of numerical range in a space C n equipped with a sesquilinear form. Li in , generalized the work of Tsing and explored fundamental properties and consequences of numerical range in the framework sesquilinear form. We also refer to another interesting paper by Fox  of this type. ### Orlicz norm inequalities for the composite operator and applications

Differential forms as the extensions of functions have been rapidly developed. In recent years, some important results have been widely used in PDEs, potential theory, non- linear elasticity theory, and so forth; see [1-7] for details. However, the study on opera- tor theory of differential forms just began in these several years and hence attracts the attention of many people. Therefore, it is necessary for further research to establish some norm inequalities for operators. The purpose of this article is to establish Orlicz norm inequalities for the composition of the homotopy operator T and the projection operator H. ### ON SOME INEQUALITIES FOR THE GENERALIZED EUCLIDEAN OPERATOR RADIUS

Abstract. There are many criterion to generalize the concept of numerical radius; one of the most re- cent interesting generalization is what so called the generalized Euclidean operator radius. Simply, it is the numerical radius of multivariable operators. In this work, several new inequalities, refinements and generalizations are established for this kind of numerical radius. ### On a Hilbert Type Operator with a Class of Homogeneous Kernels

By using the way of weight coe ﬃ cient and the theory of operators, we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of −λ-degree is established, and some particular cases are considered. ### On Norm of Elementary Operator: An Application of Stampfli’s Maximal Numerical Range

Properties of elementary operators have been investigated in the resent past under varied aspects. Their norms have been a subject of interest for research in operator theory. Deriving a formula to express the norm of an arbitrary elementary operator in terms of its coefficient operators remains a topic of research in operator theory. In the current paper, the concept of the maximal numerical range is applied in determining the lower bound of the norm of an elementary operator consisting of two terms, and also to determine the conditions under which the norm of this operator is expressible in terms of its coefficient operators in . Specifically, the Stampfli’s maximal numerical range is employed in arriving at our results. ### Further Inequalities for Sequences and Power Series of Operators in Hilbert Spaces Via Hermitian Forms

w (T ) = sup {|λ| , λ ∈ W (T )} = sup {|hT x, xi| , kxk = 1} . (46) It is well known that w (·) is a norm on the Banach algebra B (H) of all bounded linear operators T : H → H. This norm is equivalent with the operator norm. In fact, the following more precise result holds [27, p. 9]: ### Reverse Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces

Remark 1. If the operator T : H → H is such that R (T) ⊥ R (T ∗ ) , kT k = 1 and kT − Ik ≤ 1, then the equality case holds in (2.2). Indeed, by Theorem 6, we have in this case w (T ) = 1 2 kT k = 1 2 and since we can choose in Theorem 8, λ = 1, r = 1, then we get in both sides of (2.2) the same quantity 1 ### Inequalities Of Schwarz Type For n-Tuples Of Vectors In Inner Product Spaces With Applications

w (T ) = sup {|λ| , λ ∈ W (T )} = sup {|hT x, xi| , kxk = 1} . (3.2) It is well known that w (·) is a norm on the Banach algebra B (H ) of all bounded linear operators on the Hilbert space H. This norm is equivalent with the operator norm. In fact, the following more precise result holds [17, p. 9]: