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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]

In this paper, we present various inequalities regarding the linear combinations of orthogonal projections. These results aim to generalize and refine well-known inequalities, such as those due to Buzano and Ostrowski. Additionally, we investigate a specific case of these linear combinations and introduce new refinements of the Cauchy–Schwarz inequality. Furthermore, we establish some findings related to the covariance and variance of bounded linear operators. Moreover, as applications of some of our results, we establish several inequalities involving the product of three operators, one of which is a linear combination of an orthogonal projection and the identity operator. Finally, we introduce a new positive operator construction in terms of an orthogonal projection and the identity operator, and we derive some norms and numerical radius inequalities involving it. [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]

Let A be a positive (semidefinite) bounded linear operator on a complex Hilbert space (H , ⟨ · , · ⟩) . The semi-inner product induced by A is defined by ⟨ x , y ⟩ A : = ⟨ A x , y ⟩ for all x , y ∈ H and defines a seminorm ‖ · ‖ A on H . This makes H into a semi-Hilbert space. For p ∈ [ 1 , + ∞) , the generalized A-joint numerical radius of a d-tuple of operators T = (T 1 , ... , T d) is given by ω A , p (T) = sup ‖ x ‖ A = 1 ∑ k = 1 d | 〈 T k x , x 〉 A | p 1 p. Our aim in this paper is to establish several bounds involving ω A , p (·) . In particular, under suitable conditions on the operators tuple T , we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005). [ABSTRACT FROM AUTHOR]

This paper introduces a new family of non-negative real-valued functions on the algebra of all bounded linear operators on a Hilbert space. These functions will be called semi-norms because they satisfy the norm axioms, except for the triangle inequality, which will be discussed for special cases. Many bounds and relations will be shown for this new family, with a connection to the existing 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]

In this paper, we study the norm and skew angular distances in a normed space X , where convex functions are used to obtain refinements and reverses of some outstanding results in the literature. For example, in this regard, we show that if a , b ∈ X are non-zero and if p , q > 0 are such that 1 p + 1 q = 1 , then 2 λ p r a r + q r b r 2 - p a + q b 2 r ≤ p r - 1 a r + q r - 1 b r - a + b r ≤ 2 μ p r a r + q r b r 2 - p a + q b 2 r , where r ≥ 1 , λ = min 1 / p , 1 / q and μ = max 1 / p , 1 / q . Then we explain how this result extends some known results in the literature. Many other related results will be also shown. Then, with the theme of convexity, we employ a log-convex approach on certain matrix functions to obtain improvements and new sights of some matrix inequalities, including possible bounds of ‖ A t X B 1 - t ‖ , where A, B are positive definite matrices, X is an arbitrary matrix, ‖ · ‖ is a unitarily invariant norm and 0 ≤ t ≤ 1. Many other results involving matrix and scalar log-convex functions will be presented too. [ABSTRACT FROM AUTHOR]

In this paper, we establish several characterizations of the A-parallelism of bounded linear operators with respect to the seminorm induced by a positive operator A acting on a complex Hilbert space. Among other things, we investigate the relationship between A-seminorm-parallelism and A-Birkhoff–James orthogonality of A-bounded operators. In particular, we characterize A-bounded operators which satisfy the A-Daugavet equation. In addition, we relate the A-Birkhoff–James orthogonality of operators to the distance formulas and we give an explicit formula of the center mass for A-bounded operators. Some other related results are also discussed. [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]

Let H be a Hilbert space, $$ \dim H = \infty $$ . The set Δ1 = {1 + a : a in the trace class, 1 + a positive and invertible} is a differentiable manifold of operators, and a homogeneous space under the action of the invertible operators g which are themselves nuclear perturbations of the identity (one of the called classical Banach-Lie groups): In this paper we introduce a Finsler metric in Δ1, which is invariant under the action. We investigate the metric space thus induced. For instance, we prove that it is complete non-positively curved (in the sense of Busemann). Other geometric properties are derived. [ABSTRACT FROM AUTHOR]