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In this paper, we explore various transforms associated with a bounded linear operator T on a Hilbert space. These transforms include the Aluthge, λ -Aluthge, Duggal, generalized mean, and λ -mean transforms. Our aim is to investigate the connections between T and these transforms, focusing on aspects such as norm inequalities and numerical ranges, while also highlighting certain essential properties. Furthermore, we aim to determine the conditions under which an operator T coincides in norm with its transformed counterparts through these transformations. Several characterizations and properties are also derived. [ABSTRACT FROM AUTHOR]

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set Θ ⊆ C such that the evaluation functionals φ τ (f) = f (τ) , τ ∈ Θ , are continuous on ℋ. The Berezin number of an operator X is defined by ber (X) = sup τ ∈ Θ | X ˜ (τ) | = sup τ ∈ Θ | 〈 X k ˆ τ , k ˆ τ 〉 | , where the operator X acts on the reproducing kernel Hilbert space H = H (Θ) over some (non-empty) set Θ . In this paper, we introduce a new family involving means ∥ ⋅ ∥ σ t between the Berezin radius and the Berezin norm. Among other results, it is shown that if X ∈ L (H) and f , g are two non-negative continuous functions defined on [ 0 , ∞) such that f (t) g (t) = t , (t ⩾ 0) , then ∥ X ∥ σ 2 ⩽ ber (1 4 (f 4 (| X |) + g 4 (| X ∗ |)) + 1 2 | X | 2) and ∥ X ∥ σ 2 ⩽ 1 2 ber (f 4 (| X |) + g 2 (| X | 2)) ber (f 2 (| X | 2) + g 4 (| X ∗ |)) , where σ is a mean dominated by the arithmetic mean ∇ . [ABSTRACT FROM AUTHOR]

Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius. [ABSTRACT FROM AUTHOR]

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set Θ ⊆ C such that the evaluation functionals φ τ (f) = f (τ) , τ ∈ Θ , are continuous on ℋ. The Berezin number of an operator X is defined by ber (X) = sup τ ∈ Θ | X ˜ (τ) | = sup τ ∈ Θ | 〈 X k ˆ τ , k ˆ τ 〉 | , where the operator X acts on the reproducing kernel Hilbert space H = H (Θ) over some (non-empty) set Θ . In this paper, we introduce a new family involving means ∥ ⋅ ∥ σ t between the Berezin radius and the Berezin norm. Among other results, it is shown that if X ∈ L (H) and f , g are two non-negative continuous functions defined on [ 0 , ∞) such that f (t) g (t) = t , (t ⩾ 0) , then ∥ X ∥ σ 2 ⩽ ber (1 4 (f 4 (| X |) + g 4 (| X ∗ |)) + 1 2 | X | 2) and ∥ X ∥ σ 2 ⩽ 1 2 ber (f 4 (| X |) + g 2 (| X | 2)) ber (f 2 (| X | 2) + g 4 (| X ∗ |)) , where σ is a mean dominated by the arithmetic mean ∇ . [ABSTRACT FROM AUTHOR]

In this article, we delve into the classic Bohr inequality for complex numbers, a fundamental result in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr's inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, as well as leveraging the class of functions defined by the Daykin–Eliezer–Carlitz inequality. Our novel contribution lies in demonstrating that Bohr's and Bergström's inequalities can be derived from one another, revealing a deeper interconnectedness between these results. Furthermore, we present several new generalizations of Bohr's inequality, along with other notable inequalities from the literature, and discuss their various implications. By providing more comprehensive and verifiable conditions, our work extends previous research and enhances the understanding and applicability of Bohr's inequality in mathematical studies. [ABSTRACT FROM AUTHOR]

The classical Bohr inequality for scalars was extended to the non-commutative case of Hilbert space operators in the literature. The sole goal of this article is to discuss the operator Bohr inequality and present some of its new variants. This includes fresh reverses and refinements of this inequality with applications towards an operator's real and imaginary parts, not to forget the new discussion of different domains of the parameters. One further application towards the operator Dunkl-Williams inequality will be presented too. While the new results are interesting, we emphasize that the approach used to explore these inequalities differs from the existing literature methods for this context. [ABSTRACT FROM AUTHOR]

A simple convex approach and block techniques are used to obtain new sharpened versions of numerical radius inequalities for Hilbert space operators. These include comparisons of norms of operators, their Cartesian parts, their numerical radii, and the numerical radius of the product of two operators. [ABSTRACT FROM AUTHOR]

The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers several bounds involving the summation of operators, notably the Selberg operator. Through the examination of these properties and relationships, this study contributes to a better understanding of the Selberg operator and its influence on operator compositions. The paper also highlights the significance of symmetry in mathematics and its potential implications across various mathematical domains. [ABSTRACT FROM AUTHOR]

Our aim in this article is to establish several new norm and numerical radius inequalities for products and sums of operators acting on a complex Hilbert space (H , 〈 · , · 〉) . Some of the obtained inequalities improve well known ones. In addition, by using new techniques, we prove certain new inequalities related to ω A (·) and | | · | | A , where ω A (T) and | | T | | A denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbert space (H , 〈 · , · 〉 A) , respectively. Here 〈 x , y 〉 A : = 〈 A x , y 〉 for every x , y ∈ H . [ABSTRACT FROM AUTHOR]

This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg operator, we derive improved versions of the classical Selberg inequality. Additionally, we demonstrate the interdependence among well-known inequalities such as Cauchy–Schwarz, Bessel, and the Selberg inequality, revealing that these inequalities can be deduced from one another. This study showcases the enhancements made to the classical Selberg inequality and establishes the interconnectedness of various mathematical inequalities. [ABSTRACT FROM AUTHOR]

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space H , which are bounded with respect to the seminorm induced by a positive operator A on H . Here A is not assumed to be invertible. Mainly, if we denote by ω A (⋅) and ω (⋅) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ω A (T) = ω ( A 1 / 2 T (A 1 / 2 ) † ) , where (A 1 / 2 ) † is the Moore-Penrose inverse of A 1 / 2 . In addition, several new inequalities involving ω A (⋅) for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496]. [ABSTRACT FROM AUTHOR]

Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family of pairs of noncospectral singularly cospectral graphs with unbounded number of vertices. It is clear that almost cospectral graphs are also singularly cospectral but the converse is not necessarily true, we present families of graphs where both concepts: almost cospectrality and singularly cospectrality agree. [ABSTRACT FROM AUTHOR]

Let B (n , q) be the class of block graphs on n vertices having all their blocks of the same size. We prove that if G ∈ B (n , q) has at most three pairwise adjacent cut vertices then the minimum spectral radius ρ (G) is attained at a unique graph. In addition, we present a lower bound for ρ (G) when G ∈ B (n , q). [ABSTRACT FROM AUTHOR]

Let G (n , α) be the class of block graphs on n vertices and prescribed independence number α. In this article we prove that the maximum spectral radius ρ (G) , among all graphs G ∈ G (n , α) , is reached at a unique graph. As a byproduct we obtain an upper for ρ (G) , when G ∈ G (n , α). [ABSTRACT FROM AUTHOR]

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. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some refinements of the known operator inequalities for the p -Schatten class. In addition, we obtain an approach to the inequalities conjectured by Audenaert and Kittaneh for the p -Schatten class. [ABSTRACT FROM AUTHOR]

In this paper we present refinements and improvement of the Young inequality in the context of linear bounded operators [ABSTRACT FROM AUTHOR]

Abstract: Sampling theory is concerned with the problem of reconstructing a signal in a Hilbert space from a given a collection of sampled values of . If a certain decomposition of the Hilbert space is possible (in terms of the sampling and reconstruction subspaces) then a consistent reconstruction can be obtained. In this paper we treat the case in which such a decomposition cannot be found. For this situation, we study the quasi-consistent reconstructions which are an extension of the consistent reconstructions. We relate the previous concepts to generalized inverses. We also present some new results and problems regarding consistent sampling. [Copyright &y& Elsevier]

Abstract: We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the Corach–Porta–Recht inequality. We also characterize the class of operators satisfying under certain conditions. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a family of examples that can be regarded as spaces of nonpositive curvature, but with the distinct quality that they are not complete as metric spaces. This amounts to the fact that they are modelled on a finite von Neumann algebra, and the metrics introduced arise from the trace of the algebra. In spite of the noncompleteness of these manifolds, their geometry can be studied from the view-point of metric geometry, and several techniques derived from the functional analysis are applied to gain insight on their geodesic structure. [Copyright &y& Elsevier]

Abstract: In this work we characterize normal invertible operators via inequalities with unitarily invariant norm of elementary operators. [Copyright &y& Elsevier]

Abstract: We study the geometry of the set , with , which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of . These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920–931], we introduced the metric and exposed several results about this metric space. The aim of this work is to prove that the space behaves in many senses like a nonpositive curvature metric space. [Copyright &y& Elsevier]

Abstract: The aim of this work is to apply the complex interpolation method to norms of -tuples of operators in a symmetrically-normed ideal defined by a symmetric norming function (s.n.f.). The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted -norms, the weight being a positive invertible operator. [Copyright &y& Elsevier]

Abstract: The aim of this work is to apply the complex interpolation method to norms of n-tuples of operators in the p-Schatten classes of a Hilbert space H. The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted p-norms, the weight being a positive invertible operator. As a by-product, we obtain Clarkson''s type inequalities. [Copyright &y& Elsevier]

Abstract: The celebrated Heinz inequality asserts that for , , every unitarily invariant norm and . In this paper, we present several improvement of the Heinz inequality by using the convexity of the function , some integration techniques and various refinements of the Hermite–Hadamard inequality. In the setting of matrices we prove that for real numbers α, β. [Copyright &y& Elsevier]