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.

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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.

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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.

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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),

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 [9], Dragomir-S´ andor [7] and Dragomir [5].

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.

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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.

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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 [14] [15]. In [16], 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 [17] [18]. 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.

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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

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.

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

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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.

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During the past decades, several definitions of the **numerical** range in various settings have been introduced by many mathematicians. For instance, Marcus and Wang [15] opened the concept of the rth permanent **numerical** range of **operator** A. Furthermore, Descloux in [3] defined the notion of the essential **numerical** range of an **operator** with respect to a coercive sesquilinear form. In 1977, Marvin [16] and in 1984, independently, Tsing [23] introduce and characterize a new version of **numerical** range in a space C n equipped with a sesquilinear form. Li in [14], 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 [10] of this type.

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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.

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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**.

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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.

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.

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]:

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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

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]:

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