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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 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]

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]

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]

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]

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]

This paper studies the metric structure of manifolds of semi-negative curvature. Explicit estimates on the geodesic distance and sectional curvature are obtained in the setting of homogeneous spaces G/K of Banach–Lie groups, and a characterization of convex homogeneous submanifolds is given in terms of the Banach–Lie algebras. A splitting theorem via convex expansive submanifolds is proved, inducing the corresponding splitting of the Banach–Lie group G. The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces, and manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class of nonpositively curved spaces. Several well-known results, such as the existence and uniqueness of best approximations from convex closed sets, or the Bruhat–Tits fixed-point theorem, are shown to hold in this setting, without dimension restrictions. Finally, these notions are used to study the structure of the classical Banach–Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such a setting. [ABSTRACT FROM PUBLISHER]

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]