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Assume that h : G → ℂ {h:G\rightarrow\mathbb{C}} is analytic on the convex domain G and x ∈ ℒ ⁢ ( ℬ ; E , 풜 , μ ) {x\in\mathcal{L}(\mathcal{B};E,\mathcal{A},\mu)} , the set of Bochner-integrable functions on a measurable space ( E , 풜 , μ ) {(E,\mathcal{A},\mu)} endowed with a countably-additive scalar measure μ on a σ-algebra 풜 {\mathcal{A}} of subsets of E and with values in the Banach algebra ℬ {\mathcal{B}} . If the spectrum σ ⁢ ( x ⁢ ( t ) ) ⊂ G {\sigma(x(t))\subset G} for all t ∈ E {t\in E} and γ ⊂ G {\gamma\subset G} is taken to be close rectifiable curve in G such that σ ⁢ ( x ⁢ ( t ) ) ⊂ ins ⁡ ( γ ) {\sigma(x(t))\subset\operatorname*{ins}(\gamma)} for all t ∈ E {t\in E} , then, in this paper, we show among others that ∥ ∫ E ( h ∘ x ) ⁢ ( u ) ⁢ 푑 μ ⁢ ( u ) - h ⁢ ( ∫ E x ⁢ ( u ) ⁢ 푑 μ ⁢ ( u ) ) ∥ ≤ 1 2 ⁢ π ⁢ ∫ E ∥ x ¯ E - x ⁢ ( v ) ∥ ⁢ ( ∫ γ | h ⁢ ( ξ ) | ( | ξ | - ∥ x ¯ E ∥ ) ⁢ ( | ξ | - ∥ x ⁢ ( v ) ∥ ) ⁢ | d ⁢ ξ | ) ⁢ 푑 μ ⁢ ( v ) , \Bigg{\|}\int_{E}(h\circ x)(u)\,d\mu(u)-h\Bigg{(}\int_{E}x(u)\,d\mu(u)\Bigg{)}% \Bigg{\|}\leq\frac{1}{2\pi}\int_{E}\|\bar{x}_{E}-x(v)\|\Bigg{(}\int_{\gamma}% \frac{|h(\xi)|}{(|\xi|-\|\bar{x}_{E}\|)(|\xi|-\|x(v)\|)}|\,d\xi|\Bigg{)}\,d\mu% (v), where x ¯ E := ∫ E x ⁢ ( u ) ⁢ 푑 μ ⁢ ( u ) . \bar{x}_{E}:=\int_{E}x(u)\,d\mu(u). Some examples for exponential function in Banach algebras are also given. Applications for discrete inequalities and Hermite–Hadamard-type inequalities are provided as well. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to provide a full asymptotic analysis of a family of neural network (NN) operators based on suitable density functions within the L p -setting, and in the space of continuous functions. Two approaches are pursued: the first employs the celebrated Hardy–Littlewood (HL) maximal inequality, while the second adopts a constructive, fully moment-based, method. A crucial step in the proof of the previous results is provided by achieving asymptotic estimates for the NN operators in the cases of functions belonging to Sobolev spaces. By means of the previously mentioned first approach, we are able to establish sharp estimates that can not be applied with p = 1 , since in that case, the HL maximal inequality fails. This justifies resorting to the second complementary approach, which is revealed to be very useful to cover the remaining case. The asymptotic analysis is finally completed by deducing the corresponding qualitative order of approximation for functions within suitable Lipschitz classes. At the end of the paper, several examples of density functions are also presented and discussed in relation to the previous results. Finally, we recall that NN operators based on the well-known ReLU or RePUs functions are also included in the present theory. [ABSTRACT FROM AUTHOR]

The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n-dimensional balls, valid for the largest variety of underlying parameters and for all dimensions n ∈ N , n ≥ 2 . [ABSTRACT FROM AUTHOR]

Let A, B, X, and Y be n × n complex matrices such that A is self-adjoint, B ≥ 0 , ± A ≤ B , max (∥ X ∥ 2 , ∥ Y ∥ 2) ≤ 1 , and let f be a nonnegative increasing convex function on [ 0 , ∞) satisfying f (0) = 0 . Then 2 s j (f (| X A Y ∗ |)) ≤ max { ∥ X ∥ 2 , ∥ Y ∥ 2 } s j (f (B + A) ⊕ f (B − A)) for j = 1 , 2 , ... , n . This singular value inequality extends an inequality of Audeh and Kittaneh. Several generalizations for singular value and norm inequalities of matrices are also given. [ABSTRACT FROM AUTHOR]

In this article, we present generalized improvements of certain Cauchy–Bunyakovsky–Schwarz type inequalities. As applications of our results, we provide improvements of some numerical radius inequalities for Hilbert space operators. Finally, we obtain certain numerical radii of Hilbert space operators involving geometrically convex functions. [ABSTRACT FROM AUTHOR]

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]

This study explores the power vector inequalities for a pair of operators (B,C)\left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where (B,C)\left(B,C) is equal to (A,A*)\left(A,{A}^{* }) or (Re(A),Im(A))(\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator AA in B(H)B\left(H), where HH is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the ss-rr-norm and ss-rr -numerical radius of the operator pair (B,C)∈B(H)\left(B,C)\in B\left(H), which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A∈B(H)A\in B\left(H).

Let A = A ij be an n × n operator matrix, where each A ij is a bounded linear operator on a complex Hilbert space. Among other numerical radius bounds, we show that w (A) ≤ w (A ^) , where A ^ = a ^ ij is an n × n complex matrix, with a ^ ij = w (A ii ) when i = j , | A ij | + | A ji ∗ | 1 / 2 | A ji | + | A ij ∗ | 1 / 2 when i < j , 0 when i > j. This is a considerable improvement of the existing bound w (A) ≤ w (A ~) , where A ~ = a ~ ij is an n × n complex matrix, with a ~ ij = w (A ii ) when i = j , ‖ A ij ‖ when i ≠ j. Further, applying the bounds, we develop the numerical radius bounds for the product of two operators and the commutator of operators. Also, we develop an upper bound for the spectral radius of the sum of the product of n pairs of operators, which improves the existing bound. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop a dilation theory on the regular U -twisted polyball B U (H) , which is the set of all k-tuples T : = (T 1 , … , T k) of row contractions T i : = [ T i , 1 ⋯ T i , n i ] on a Hilbert space H satisfying certain positivity condition on the defect operator Δ T (I) and U -commutation relations, where U ⊂ B (H) is an appropriate set of unitary operators. The role of operator model is played by a standard multi-shift S : = (S 1 , … , S k) with S i : = [ S i , 1 ⋯ S i , n i ] , which is a k-tuple of doubly I ⊗ U -commuting pure row isometries on the Hilbert space ℓ 2 (F n 1 + × ⋯ × F n k +) ⊗ H , where F n i + is the unital free semigroup with n i generators and each S i , j is a weighted shift with operator-valued weights. It is shown that many of the classical results concerning the dilation theory of contractions on Hilbert spaces have analogues for U -twisted polyballs. This includes: Sz.-Nagy dilation theorem, von Neumann inequality, Itô and Brehmer dilations for commuting isometries and contractions, respectively, and Beurling characterization of the invariant subspaces for the unilateral shift on the Hardy space H 2 . [ABSTRACT FROM AUTHOR]

We investigate the geometric properties for a class of trace functions expressed in terms of the deformed logarithmic and exponential functions. We extend earlier results of Epstein, Hiai, Carlen and Lieb. [ABSTRACT FROM AUTHOR]

In this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality ⊥ B ϵ . First, we show the relation between linear functionals and hyperplanes by means of approximate Birkhoff orthogonality. Second, we establish the sufficient conditions for Birkhoff orthogonality and approximate Birkhoff orthogonality. Then some sufficient and necessary conditions for isosceles orthogonality and Pythagorean orthogonality of bounded operators are shown. In addition, we characterize the inner product space in terms of functional orthogonality. [ABSTRACT FROM AUTHOR]

In this paper, we study operator mean inequalities for the weighted arithmetic, geometric, and harmonic means. We give a slight modification of Audenaert's result to show the relation between Kwong functions and operator monotone functions. Operator mean inequalities provide some analogs of the geometric concavity property for Kwong functions, operator convex, and operator monotone functions. Moreover, we give our points across by way of some examples which show the usage of our main results. [ABSTRACT FROM AUTHOR]

We compute certain inequalities for B-Berezin radius of 2 × 2 operator matrices in the study that generalize and refine earlier inequalities. Furthermore, we construct A-Berezin radius inequalities of operators in B A , Υ (H) that improve on the current inequalities in Huban (Turk J Math 46(1):189–206, 2022). In addition, we establish A-Berezin radius bounds for sum of product of operators in B A , Υ (H) , which improve on the previous bounds. [ABSTRACT FROM AUTHOR]

Let ßA (ℍ) denote the algebra of bounded linear operators on a complex Hilbert space ℍ that admit A-adjoint operators, where A is a non-zero positive semi-definite operator on ℍ. A commuting operator tuple T = (T1 ,..., Td) ∈ ßA(ℍ)d is called jointly A-normaloid if rA(T) = ∥T∥A, where rA(T) and ∥T∥A represent the joint A-spectral radius and the joint operator A-seminorm of T, respectively. This paper aims to investigate this new class of operators and provides several examples. Furthermore, a characterization of A-normaloidity is established. Additionally, the joint Euclidean A-seminorm of a d-tuple of A-bounded operators T, denoted by , is examined. Specifically, for all positive integers n, we prove that the following equivalence holds for any commuting operator tuple. Here. Finally, several related questions are explored. [ABSTRACT FROM AUTHOR]

This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.

In this note, some inequalities involving matrix means of sectorial matrices are proved which are generalizations and refinements of previous known results. Among them, let A and B be two accretive matrices with A , B ∈ S θ , 0 < m I ⩽ A , B ⩽ M I for positive real numbers M and m. If σ , σ 1 , σ 2 are matrix means such that σ ∗ ⩽ σ 1 , σ 2 ⩽ σ , where σ ∗ is the adjoint of σ and Φ is a positive unital linear map, then for each p > 0 , Φ p ℜ (A σ 1 B) ⩽ sec 2 p θ α p Φ p ℜ (A σ 2 B) , where α = max K , 4 1 - 2 p K , and K = (M + m) 2 4 m M is the Kantorovich constant. [ABSTRACT FROM AUTHOR]

We improve the generalized numerical radius version of the Heinz means inequalities by employing convexity and the Hermit-Hadamard inequality. Moreover, we present novel proofs for the generalized numerical radius of Heinz-type inequalities. [ABSTRACT FROM AUTHOR]

Let A be a unital JB-algebra and A , B ∈ A . The weighted geometric mean A ♯ r B for A , B ∈ A has been recently defined for r ∈ [ 0 , 1 ]. In this work, we extend the weighted geometric mean A ♯ r B , from r ∈ [ 0 , 1 ] to r ∈ (- 1 , 0) ∪ (1 , 2) . We will notice that many results will be reversed when the domain of r change from [0, 1] to (- 1 , 0) or (1, 2). We also introduce the Heinz and Heron means of elements in A , and extend some known inequalities involving them. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present new inequalities for the spectral geometric mean $ A\natural _t B $ A ♮ t B of two positive definite operators A, B on a Hilbert space. The obtained results complement many known inequalities for the geometric mean $ A\sharp _t B $ A ♯ t B. In particular, explicit comparisons between $ A\natural _t B $ A ♮ t B and $ A\sharp _t B $ A ♯ t B are given, with applications towards Ando-type inequalities and Ando-Hiai inequalities for $ A\natural _t B $ A ♮ t B and some other consequences. [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]

Let $ \mathcal {L}(\mathscr {H}) $ L (H) denote the $ C^* $ C ∗ -algebra of adjointable operators on a Hilbert $ C^* $ C ∗ -module $ \mathscr {H} $ H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in $ \mathscr {L}(\mathscr {H}) $ L (H). More precisely, an operator $ A\in \mathscr {L}(\mathscr {H}) $ A ∈ L (H) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists $ \nu \in (0, 1) $ ν ∈ (0 , 1) such that \[ \|\langle Ax, y\rangle\|\leq (\|Ax\|\|y\|)^{\nu}(\|Ay\|\|x\|)^{1 - \nu} \quad (x, y \in \mathscr{H}). \] ‖ ⟨ A x , y ⟩ ‖ ≤ (‖ A x ‖ ‖ y ‖) ν (‖ A y ‖ ‖ x ‖) 1 − ν (x , y ∈ H). We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if $ \left \|\langle Ax, y\rangle \right \| \leq \left \|{|A|}^{1/2}x\right \|\left \|{|A|}^{1/2}y\right \| $ ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all $ x, y \in \mathscr {H} $ x , y ∈ H . [ABSTRACT FROM AUTHOR]

The operators T on a Hilbert space H which have 3-isometric liftings S on Hilbert spaces K containing H are investigated. Here, we deal with the liftings S which are 2-isometries on their invariant subspace K ⊖ H and also have this subspace invariant for S ∗ S . Several characterizations for such operators T are obtained, including the case when the lifting S is expansive. As an application, we refer to the class of (A, 2)-contractions for a positive operator A on H , and particularly to the expansive 3-concave operators. Also, we obtain an upper triangulation for T induced by S, where the operators on the main diagonal are close to contractions. Finally, we study such liftings S which are minimal, in the sense that K is generated by { S n H , n ≥ 0 } . [ABSTRACT FROM AUTHOR]

We present several upper bounds for the numerical radii of 2 × 2 operator matrices. We employ these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then for every r ≥ 1 and α ∈ [0, 1]. This substantially improves on the existing inequality. Here w(.) and ||.|| denote the numerical radius and the usual operator norm, respectively. [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]

In this article we introduce a new relation on the cone of positive definite matrices and we study its properties and its effect on operator monotonicity and convexity. Furthermore, we use this new relation to establish analogies between the weighted geometric means A ♯ t B and the spectral weighted geometric means A ♮ t B of positive definite matrices A and B, via the Thompson metric d ∞ (A , B) and the semi-metric d (A , B) = 2 ‖ log (A - 1 ♯ B) ‖. [ABSTRACT FROM AUTHOR]

For positive definite matrices A and B, the Kubo-Ando matrix power mean is defined as \[ P_\mu(p, A, B) = A^{1/2}\left(\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\right)^{1/p} A^{1/2}\quad (p \ge 0). \] P μ (p , A , B) = A 1 / 2 (1 + (A − 1 / 2 B A − 1 / 2 ) p 2) 1 / p A 1 / 2 (p ≥ 0). In this paper, for $ 0\le p \le 1 \le q $ 0 ≤ p ≤ 1 ≤ q , we show that if one of the following inequalities \[ f(P_\mu(p, A, B)) \le f(P_\mu(1, A, B)) \le f(P_\mu(q, A, B)) \] f (P μ (p , A , B)) ≤ f (P μ (1 , A , B)) ≤ f (P μ (q , A , B)) holds for any positive definite matrices A and B, then the function f is operator monotone on $ (0, \infty). $ (0 , ∞). We also study the inverse problem for non-Kubo-Ando matrix power means with the powers 1/2 and 2. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means. [ABSTRACT FROM AUTHOR]

In this paper, we prove some operator inequalities associated with an extension of the Kantorovich type inequality for s-convex functions. We also give an application to the order preserving power inequality of three variables and find a better lower bound for the numerical radius of a positive operator under some conditions. [ABSTRACT FROM AUTHOR]

We analyze the role played by functions with positive differences defined on convex cones. In particular, we study functions that satisfy linear functional inequalities that extend the three-variable Hornich-Hlawka functional inequality, f x + f y + f z + f x + y + z ≥ f x + y + f y + z + f z + x + f (0) , especially to the case of n variables. Beyond the classical setting, we present extensions to the case of positive operators. [ABSTRACT FROM AUTHOR]

For a continuous and positive function w (λ) , λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform 𝒟(w,μ)(T):=∫0∞w(λ)(λ+T)-1dμ(λ),\mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.

Let f : (0 , ∞) → R be an operator monotone function and A ∈ M n be accretive matrix, we show that f (A) = a + b A + ∫ 0 ∞ λ A (λ + A) - 1 d μ (λ) , where b ≥ 0 , a ∈ R and μ is a positive measure on the closed positive half-line [ 0 , ∞) with ∫ λ 1 + λ d μ (λ) < ∞ . We also show that If f : (0 , ∞) → (0 , ∞) is an operator monotone function and A , B ∈ S α . Then d (R f (A) , R f (B)) ≤ 1 + 4 | log cos (α) | d (R (A) , R (B)) , where d is the Thompson distance on positive definite operators. Finally, if f is an operator convex function on (0 , ∞) , Φ is a unital positive linear map on B (H) and A ∈ B (H) with 0 < m I ≤ A ≤ M I . Then, we get K (h , 2) f (Φ (A)) ≥ Φ (f (A)) ≥ f (Φ (A)) , where h = M m . [ABSTRACT FROM AUTHOR]

We prove the information monotonicity of the operator perspective and a complementary inequality for the information monotonicity of the operator perspective involving a positive linear map. As some consequences of our results, we demonstrate the information monotonicity and its complementary for the α -power operator mean, the Tsallis relative operator entropy, the generalized Tsallis relative operator entropy, the relative operator entropy, the generalized relative operator entropy, and the weighted harmonic mean. [ABSTRACT FROM AUTHOR]

A numerical radius inequality due to Kittaneh states that if M is a bounded linear operator on a complex separable Hilbert space, then w 2 (M) ≤ 1 2 M ∗ M + M M ∗ . We prove a numerical radius inequality for finite sums of operators, which is a considerable generalization of Kittaneh inequality: Let M 1 , M 2 , … , M n , N 1 , N 2 , … , N n be operators on a complex separable Hilbert space and let f, g be nonnegative continuous functions on 0 , ∞ which are satisfying the relation f (a) g (a) = a for all a ∈ 0 , ∞. Then for r ≥ 2 , w r ∑ i = 1 n (M i + N i) ≤ 2 r - 2 Z , where Z = ∑ i = 1 n f 2 r M i + f 2 r N i + g 2 r M i ∗ + g 2 r N i ∗ . This inequality is attractive generalization of several recent inequalities. [ABSTRACT FROM AUTHOR]

New extensions of the Cauchy–Schwarz inequality in the framework of pre-Hilbert C ∗ -modules are given. An application to the numerical radius in C ∗ -algebras is also provided. [ABSTRACT FROM AUTHOR]

The bounded linear operators T on a Hilbert space which have 2-isometric liftings S on are investigated in this paper. We refer to the structure of those operators in the case of general liftings, as well as for a more special type of such liftings S, namely for which is invariant to the orthogonal projection onto the kernel of S∗S − I. In this special case we describe the canonical triangulation of T induced by S, with entries expressed by operators close to contractions or isometries. Also, in this case we refer to some asymptotic properties of T and of its adjoint T∗ obtained by means of asymptotic limits associated to T∗ and T, relative to a lifting S for T. The canonical triangulations of T in this setting are described in detail, and as an application, the Wold type decomposition of a concave operator is obtained. [ABSTRACT FROM AUTHOR]