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Let k and n be two coprime positive integers. In this paper, we show that any $ n\times n $ n × n k-circulant operator matrix $ A_{k,n} $ A k , n with the first row $ A_{1},\dots,A_{n}\in B(H) $ A 1 , ... , A n ∈ B (H) is unitarily equivalent to a generalized permutation operator matrix, an operator matrix that has at most one nonzero entry in each row and in each column. Among other applications in the subject of invertibility, eigenvalues and eigenvectors, using this unitary equivalence in the particular cases of $ k^{2}\pm 1=n $ k 2 ± 1 = n $ ({\rm mod} n) $ (mod n) , we give the following new formulas for computing the numerical radius of these operators: \[ w(A_{k,n})= \begin{cases} \frac{1}{2}\displaystyle \max_{1\leq i\leq n} \sup_{\theta \in \mathbb{R}} \left\| \begin{bmatrix} {\rm e}^{\mathrm{i}\theta}B_{i+1} & B^{*}_{ki+1}\\ B^{*}_{-ki+1} & {\rm e}^{\mathrm{i}\theta}B_{-i+1} \end{bmatrix} \right\|, & {\rm if} \ k^{2}+1\\ & \quad =n \ ({\rm mod}\,n)\\ \displaystyle\max_{1\leq i \leq n}w \left(\begin{bmatrix} 0 & B_{k(i-1)+1}\\ B_{i} & 0 \end{bmatrix} \right), & {\rm if} \ k^{2}-1\\ & \quad =n \ ({\rm mod}\,n), \end{cases} \] w (A k , n) = { 1 2 max 1 ≤ i ≤ n sup θ ∈ R ‖ [ e i θ B i + 1 B ki + 1 ∗ B − ki + 1 ∗ e i θ B − i + 1 ] ‖ , if k 2 + 1 = n (mod n) max 1 ≤ i ≤ n w ([ 0 B k (i − 1) + 1 B i 0 ]) , if k 2 − 1 = n (mod n) , where $ B_{j}=\sum _{s=1}^{n}\omega ^{(s-1)(j-1)}A_{s} $ B j = ∑ s = 1 n ω (s − 1) (j − 1) A s and $ \omega ={\rm e}^{\frac {2\pi \mathrm {i}}{n}} $ ω = e 2 π i n , the nth root of unity. [ABSTRACT FROM AUTHOR]

We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if $ A=[A_{ij}] $ A = [ A i j ] is an $ n\times n $ n × n operator matrix with $ A_{ij}\in \mathbb {B}(\mathcal {H}) $ A i j ∈ B (H) for $ i,j=1,2,\dots, n $ i , j = 1 , 2 , ... , n , then $ \|A\|_{ber} \leq \left \|\left [\|A_{ij}\|_{ber}\right ]\right \| $ ‖ A ‖ b e r ≤ ‖ [ ‖ A i j ‖ b e r ] ‖ and $ \mathbf{ber}(A) \leq w([a_{ij}]), $ b e r (A) ≤ w ([ a i j ]) , where $ a_{ii}=\mathbf{ber}(A_{ii}), $ a i i = b e r (A i i) , $ a_{ij}=\big \||A_{ij}|+|A^*_{ji}|\big \|^{{1}/{2}}_{ber} \big \||A_{ji}|+|A^*_{ij}|\big \|^{{1}/{2}}_{ber} $ a i j = ‖ | A i j | + | A j i ∗ | ‖ b e r 1 / 2 ‖ | A j i | + | A i j ∗ | ‖ b e r 1 / 2 if ij. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we present some refinements and generalizations of existing numerical radius inequalities for Hilbert space operators. Further, we obtain some upper bounds of the numerical radius. We also obtain some numerical radius inequalities for 2 × 2 operator matrices which improve on some existing results. [ABSTRACT FROM AUTHOR]

AbstractIn this work, we give some p-numerical radius inequalities for operators as well as for commutators of operators. As applications, when p = ∞, we improve some earlier numerical radius inequalities for operators. Also, we refine the triangle inequalities for the numerical radius and the operator norm. [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 employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2 p -th power with p ≥ 1 of the numerical radius of the off-diagonal operator matrix 0 A B * 0 for any bounded linear operators A and B on a complex Hilbert space H. While the general matrix is not symmetric, a special case arises when B = A * , where the matrix becomes symmetric. This symmetry plays a crucial role in the derivation of our bounds, illustrating the importance of symmetric structures in operator theory. [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]

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This paper studies matrices A in M n (C) satisfying ‖ A ‖ = min { ‖ A + B ‖ : B ∈ B } , where B is a C*-subalgebra of M n (C) and ‖ · ‖ denotes the operator norm. Such an A is called B -minimal. The necessary and sufficient conditions for A to be B -minimal are characterized, and a constructive method to obtain B -minimal normal matrices is provided. Moreover, ⨁ i = 1 k B -minimal normal matrices with anti-diagonal block form are studied. [ABSTRACT FROM AUTHOR]

We develop several Euclidean numerical radius bounds for the product of two d-tuple operators using positivity criteria of a 2 × 2 block matrix whose entries are d-tuple operators. From these bounds, by using polar decomposition of operators, we obtain Euclidean numerical radius bounds for d-tuple operators. Among many other bounds, it is shown that w e (A) ≤ 1 2 ‖ A ‖ 1 / 2 ∑ k = 1 d (| A k | + | A k ∗ |) , where w e (A) and ‖ A ‖ are the Euclidean numerical radius and the Euclidean operator norm, respectively, of a d-tuple operator A = (A 1 , A 2 , ... , A d). Further, we develop an upper bound for the Euclidean numerical radius of an n × n operator matrix whose entries are d-tuple operators. In particular, it is proved that if [ A ij ] n × n is an n × n operator matrix then w e ([ A ij ] n × n) ≤ w ([ a ij ] n × n) , where each A ij is a d-tuple operator, 1 ≤ i , j ≤ n , a ij = w e (A ij) if i = j , a ij = w e (| A ji | + | A ij ∗ |) w e (| A ij | + | A ji ∗ |) if i < j , and a ij = 0 if i > j . Some related applications are also discussed. [ABSTRACT FROM AUTHOR]

We study Euclidean operator radius inequalities of d-tuple operators as well as the sum and the product of d-tuple operators. A power inequality for the Euclidean operator radius of d-tuple operators is also studied. Further, we study the Euclidean operator radius inequalities of 2 × 2 operator matrices whose entries are d-tuple operators. [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]

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]

For each pair of bounded linear operators A and B on a Hilbert space H , let ‖ A − B ‖ be their operator distance. Given an idempotent Q on H , consider the operator distances from Q to all of the projections on H. It is proved that there always exists a projection whose operator distance away from Q takes the maximum value. An example is constructed to show that such a projection may fail to be unique. [ABSTRACT FROM AUTHOR]

This paper presents some inner product inequalities for Hilbert space operators having closed ranges. The obtained results are applied to obtain new bounds for the numerical radius and the operator norm, where the Moore-Penrose inverse plays a keen role. [ABSTRACT FROM AUTHOR]

We give several lower and upper bounds for the Euclidean operator radius of two operators on a Hilbert space. We improve some earlier related bounds. Also, as applications of these bounds, we deduce some new bounds for the classical numerical radius. Some of these bounds are refinements of certain existing bounds. [ABSTRACT FROM AUTHOR]

This study deals with the one-parameter family {Dq}q∈[0,1]{\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit qq-Durrmeyer operators. The continuity of this family with respect to the parameter qq is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that {Dq}q∈[0,1]{\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q∈[0,1]q\in \left[0,1]. When it comes to the uniform operator topology, the continuity is preserved solely at q=0q=0 and fails at all q∈(0,1].q\in \left(0,1]. In addition, a few estimates for the distance between two limit qq-Durrmeyer operators have been derived in the operator norm on C[0,1]C\left[0,1].

The Cesàro means of Taylor polynomials σ n , n ≥ 0 , are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polynomials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces D ζ , they serve as a proper LPAS. The primary objective of this note is to accurately determine the norm of σ n when it is considered as an operator on D ζ. There exist several practical methods to impose a norm on D ζ , and each norm results in a distinct operator norm for σ n. In this context, we explore three different norms on D ζ and, for each norm, precisely compute the value of ‖ σ n ‖ D ζ → D ζ . Furthermore, in all instances, we identify the maximizing functions and demonstrate their uniqueness. [ABSTRACT FROM AUTHOR]

We present an optimal upper bound for the operator norm of a sectorial Hilbert space operator in terms of its numerical radius. [ABSTRACT FROM AUTHOR]

We introduce the general polynomial differentiation composition operator Tφ→,ψ→nf(z)=∑j=0nψj(z)f(j)φj(z),$$ {T}_{\overrightarrow{\varphi},\overrightarrow{\psi}}&#x0005E;nf(z)&#x0003D;\sum \limits_{j&#x0003D;0}&#x0005E;n{\psi}_j(z){f}&#x0005E;{(j)}\left({\varphi}_j(z)\right), $$where n∈ℕ0$$ n\in {\mathrm{\mathbb{N}}}_0 $$, ψj$$ {\psi}_j $$, j=0,n‾$$ j&#x0003D;\overline{0,n} $$, are holomorphic functions on the open unit disk 픻 and φj$$ {\varphi}_j $$, j=0,n‾$$ j&#x0003D;\overline{0,n} $$, are holomorphic self‐maps of 픻, calculate norm of the operator acting from the space of Cauchy transforms to the m$$ m $$th weighted‐type space, and characterize its boundedness, as well as the boundedness of the operator acting from the space of Cauchy transforms to the little m$$ m $$th weighted‐type space. [ABSTRACT FROM AUTHOR]

For m , n ∈ N , let X = (X ij) i ≤ m , j ≤ n be a random matrix, A = (a ij) i ≤ m , j ≤ n a real deterministic matrix, and X A = (a ij X ij) i ≤ m , j ≤ n the corresponding structured random matrix. We study the expected operator norm of X A considered as a random operator between ℓ p n and ℓ q m for 1 ≤ p , q ≤ ∞ . We prove optimal bounds up to logarithmic terms when the underlying random matrix X has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero ψ r ( r ∈ (0 , 2 ] ) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products A ∘ A and (A ∘ A) T . [ABSTRACT FROM AUTHOR]

This paper investigates possible upper and lower bounds for the numerical radii of 2 × 2 operator matrices. The obtained results improve and generalize many results from the literature in accessible forms. Moreover, many numerical examples will be given to support the feasibility of our findings. [ABSTRACT FROM AUTHOR]

For a large class of operators acting between weighted ℓ ∞ spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of nonnegative sequences; nonnegative, nonincreasing sequences; and nonnegative, nondecreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Cesàro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted ℓ ∞ . As an application, best constants are given for inequalities relating the weighted ℓ ∞ norms of the Cesàro and Copson operators both for general weights and for power weights. [ABSTRACT FROM AUTHOR]

Consider the power series with complex coefficients h (z) = ∑ k = 0 ∞ a k z k and its modified version h a (z) = ∑ k = 0 ∞ | a k | z k . In this article, we explore the application of certain Hölder-type inequalities for deriving various inequalities for operators acting on the aforementioned power series. We establish these inequalities under the assumption of the convergence of h (z) on the open disk D (0 , ρ) , where ρ denotes the radius of convergence. Additionally, we investigate the norm and numerical radius inequalities associated with these concepts. [ABSTRACT FROM AUTHOR]

We prove several norm and numerical radius inequalities for linear operators in Hilbert spaces. In particular, it is proved that if A is a bounded linear operator on a complex Hilbert space, then ω² (A) ≤ 1/2 ω (|A∗|A|)+||A||²), where ω (A), ||A||, and |A| are the numerical radius, the usual operator norm, and the absolute value of A, respectively [ABSTRACT FROM AUTHOR]

Let (gn)n⩾l be a sequence of independent and identically distributed positive random d × d matrices and consider the matrix product Gn = gn⋯ g1. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and Cramér-type moderate deviation expansions, for any matrix norm ∥Gn∥ of Gn, its entries G n i , j and its spectral radius ρ(Gn). Extended versions of their joint law with the direction of the random walk Gnx are also established, where x is a starting point in the unit sphere of ℝd. [ABSTRACT FROM AUTHOR]

We study the real and imaginary parts of the powers of the Volterra operator on L 2 [ 0 , 1 ] , specifically their eigenvalues, their norms and their numerical ranges. [ABSTRACT FROM AUTHOR]

We consider a metric describing the difference between real (noisy) and ideal processes that is based on the operator norm of the maximum deviation between the final real and ideal states of a quantum system. We discuss the properties as well as geometric and experimental interpretations of the metric. [ABSTRACT FROM AUTHOR]

For composite quantum systems, we consider quantum channels that uniquely determine the channels of the subsystems. Such channels of composite systems are called generating channels. Examples of generating channels are given by tensor products of two quantum channels of subsystems and their convex combinations. The paper deals with the properties of generating channels. In particular, we show that these channels form a convex compact set in the norm topology. We prove a criterion for a quantum channel to be generating. For composite systems consisting of two qubits, we construct generating phase-damping channels. For subsystems, these channels generate both phase-damping channels and depolarizing channels. Examples of nongenerating phase-damping channels are also presented. [ABSTRACT FROM AUTHOR]

This paper intends to show several inner product inequalities using the Cartesian decomposition of the operator. We utilize the obtained results to get norm and numerical radius inequalities. Our results extend and improve some earlier inequalities. Among other inequalities, it is revealed that if T is a n × n complex matrix with the imaginary part ℑT = T--T*/2i, then 1/2 max(∥TT* -- iℑT²∥1/2, ∥T*T + iℑT²∥1/2) ≤ ω(T) which is a significant improvement of the classical inequality 1/2 ℑTℑ ≤ ω(T). [ABSTRACT FROM AUTHOR]

In this paper, we give several numerical radius inequalities for Hilbert space operators by using the Young-type inequalities and the generalization of the Buzano of the Schwarz inequality. These inequalities improve the classical numerical radius inequalities. We also give the upper bound of the numerical radius of 2x2 operator matrices. [ABSTRACT FROM AUTHOR]

The geometry of the compact convex set of all n×nn\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓnp{\ell }_{n}^{p} to ℓnp{\ell }_{n}^{p} and the Schatten pp-norms, both for the range 1≤p≤∞1\le p\le \infty , have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.

Super-homogeneous functions including homogeneous functions, quasi-homogeneous functions, and several non-homogeneous functions are considered. Using the weight function method, the construction conditions of Hilbert-type multiple integral inequalities with super-homogeneous kernels are first discussed. Then, using the obtained results, the construction problem of bounded multiple integral operators with super-homogeneous kernels in weighted Lebesgue space is discussed, and the necessary and sufficient conditions for operator boundedness and the operator norm formula are obtained.

In this paper, we employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2p-th power with p≥1 of the numerical radius of the off-diagonal operator matrix 0AB*0 for any bounded linear operators A and B on a complex Hilbert space H. While the general matrix is not symmetric, a special case arises when B=A*, where the matrix becomes symmetric. This symmetry plays a crucial role in the derivation of our bounds, illustrating the importance of symmetric structures in operator theory.

Using the polar decomposition of a bounded linear operator A defined on a complex Hilbert space, we obtain several numerical radius inequalities of the operator A , which generalize and improve the earlier related ones. Among other bounds, we show that if w (A) is the numerical radius of A , then w (A) ≤ 1 2 ‖ A ‖ 1 / 2 ‖ | A | t + | A ⁎ | 1 − t ‖ , for all t ∈ [ 0 , 1 ]. Also, we obtain some upper bounds for the numerical radius involving the spectral radius and the Aluthge transform of operators. It is shown that w (A) ≤ ‖ A ‖ 1 / 2 (1 2 ‖ | A | + | A ⁎ | 2 ‖ + 1 2 ‖ A ˜ ‖) 1 / 2 , where A ˜ = | A | 1 / 2 U | A | 1 / 2 is the Aluthge transform of A and A = U | A | is the polar decomposition of A. Other related results are also provided. [ABSTRACT FROM AUTHOR]

In this work, we give some new p-numerical radius inequalities for 2 × 2 operator matrices. [ABSTRACT FROM AUTHOR]

Firstly, Hilbert-type integral inequalities with best constant factors are established for two non-homogeneous kernels. Then, by utilizing the relationship between the Hilbert-type inequality and the integral operator of same kernel, the parameter conditions for two integral operators with non-homogeneous kernels in weighted Lebesgue space to be bounded and the formula for calculating the operator norm are obtained. [ABSTRACT FROM AUTHOR]

Convergence of the differences between operator norm and spectral radius, operator norm and numerical radius, numerical radius and spectral radius, operator norm and Crawford number, operator norm and subspectral radius of complex Hilbert space operator sequences (which are uniformly convergent) has been investigated. Also, an inequality for the difference of Crawford numbers of two linear bounded operators A and B has been obtained. It is shown that c (A) - c (B) ≤ ω (A + e i θ B) for any θ ∈ R , where c (·) and ω (·) denote the Crawford number and the numerical radius, respectively. The results have been supported by an example. Finally, some applications to operator Hölder functions and operator-functions have been given. [ABSTRACT FROM AUTHOR]

In the present paper, we characterize certain compact and matrix operators on the Fibonacci spaces F θ q , studied by Gökçe and Sarıgöl (Kragujevac J. Math. 44(2), 273–286 (2020)), together with their norms and identities or estimates for the Hausdorff measures of noncompactness. [ABSTRACT FROM AUTHOR]

In this work, we give several upper bounds for the numerical radius of the sum of two operators as well as for the numerical radii of 2 × 2 operator matrices. These upper bounds refine some earlier existing ones. [ABSTRACT FROM AUTHOR]

In the present paper, by estimating operator norms, we give some characterizations of infinite matrix classes $\left( \left\vert E_{\mu }^{r}\right\vert _{q},\Lambda\right) $ and $\left( \left\vert E_{\mu }^{r}\right\vert _{\infty },\Lambda\right) $, where the absolute spaces $\ \left\vert E_{\mu }^{r}\right\vert _{q},$ $\left\vert E_{\mu }^{r}\right\vert _{\infty }$ have been recently studied by G\"{o}k\c{c}e and Sar{\i }g\"{o}l \cite{GS2019c} and $\Lambda$ is one of the well-known spaces $c_{0},c,l_{\infty },l_{q}(q\geq 1)$. Also, we obtain necessary and sufficient conditions for each matrix in these classes to be compact establishing their identities or estimates for the Hausdorff measures of noncompactness.

We develop upper and lower bounds for the numerical radius of 2 × 2 off-diagonal operator matrices, which generalize and improve on some existing ones. We also show that if A is a bounded linear operator on a complex Hilbert space, then for all r ≥ 1 , w 2 r (A) ≤ 1 4 ‖ | A | 2 r + | A ∗ | 2 r ‖ + 1 2 min ‖ ℜ (| A | r | A ∗ | r) ‖ , w r (A 2) where w(A), ‖ A ‖ and ℜ (A) , respectively, stand for the numerical radius, the operator norm and the real part of A. This (for r = 1 ) improves on some existing well-known numerical radius inequalities. [ABSTRACT FROM AUTHOR]

We give some new bounds for the numerical radii of the off-diagonal parts of 2 × 2 operator matrices. A refinement of the triangle inequality for the operator norm is also given. [ABSTRACT FROM AUTHOR]