Advances in Operator Theory and Inequalities

This study examines the lower and upper bounds of the numerical radius and the Crawford number functions of bounded linear operators on Banach spaces. It also explores some convergence properties of these functions for sequences of uniformly converging Banach space operators. Later on, similar problems are addressed in the case of sectorial operators.

Keywords: Numerical radius, Crawford number, Sectorial operator

Open Problem: Determining the numerical radii of Banach space with constant operator coefficients of abstract linear functional delay differential-operator equations plays a crucial role in establishing asymptotic stability of these equations on the infinitive interval in the Theory of Boundary Value Problems.

References
[1] F. L. Bauer, On the field of values subordinate to a norm, Numerische Mathematik 4 (1962), 103-111.
[2] G. Lumer, Semi-inner-product spaces, Transactions of the American Mathematical Society 100 (1961), 29-43.
[3] F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras. London Mathematical Society Lecture Note Series 2, Cambridge University Press, London-New York, (1971).
[4] F. F. Bonsall and J. Duncan, Numerical ranges II, London Mathematical Society Lecture Note Series 10, Cambridge University Press, New York-London, (1973).
[5] M. Martin, Numerical index theory, University of Granada, Mini-Course, Spain, (2012).
[6] A. Mal, On joint numerical radius of operators and joint numerical index of a Banach space, Operators and Matrices 17(3) (2023), 839-856.
[7] Y. Chen and Y. Wei, Numerical radius for the asymptotic stability of delay differential equations, Linear and Multilinear Algebra 65(11) (2017), 2306-2315.
[8] K. He and J. C. Hou, Applying the theory of numerical radius of operators to obtain multi-observable quantum uncertainty relations, Acta Mathematica Sinica, English Series 38(7) (2022), 1241-1254.

Our aim in this work is to give new inequalities of the d-tuples
of operators on a complex Hilbert space. An extension of the generalized some inequalities are proved for multivarible operators on a complex Hilbert space. Based on that some numerical radius inequalities due for a single operator.

Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operators on $\mathcal{H}$. For every $T\in \mathcal{B}(\mathcal{H})$, we denote by $T^{\ast}$ the adjoint of $T$. An operator $T\in \mathcal{B}(\mathcal{H})$ is said to be m-symmetric if
$$\sum_{k=0}^m (-1)^{k}\left(\begin{array}{l} m \\ k \end{array}\right)T^{\ast (m-k)}T^{k}=0,$$ where $\left(\begin{array}{l}
m \
k
\end{array}\right) $ is the binomial coefficient. $T\in\mathcal{B}(\mathcal{H})$ is called skew m-symmetric if $$\sum_{k=0}^m \left(\begin{array}{l} m \\ k \end{array}\right)T^{\ast (m-k)}T^{k}=0.$$ In this paper, we investigate in the products and the sums of m-symmetric and skew m-symmetric operators. More precisely, we show that if $T,S\in\mathcal{B}(\mathcal{H})$ are commuting operators such that T is m-symmetric and S is n-symmetric, then TS and T+S are (m+n-1)-symmetric. Furthermore, if $T,S\in\mathcal{B}(\mathcal{H})$ are commuting operators such that T is skew m-symmetric and S is skew n-symmetric, then TS is (m+n-1)-symmetric, and T+S is skew (m+n-1). Additionally, if $T,S\in\mathcal{B}(\mathcal{H})$ are commuting operators such that T is m-symmetric and S is skew n-symmetric, then TS is skew (m+n-1)-symmetric.

The mixed Cauchy-Schwarz inequality is improved to prove several Berezin radius inequalities. Berezin radius inequalities are obtained by using the doubly convex function. Furuta's inequality and with a generalization of mixed Cauchy-Schwarz inequality, demonstrates Berezin radius inequalities.

In this talk, we give an overview of the power inequality for the numerical radius, which has been known for so long. Then, we present a some new progress related to this important inequality.

In this paper, we give several p-numerical radius inequalities for
products of Hilbert space operators. For the particular case, we reobtain some earlier existing bounds. Also, we provide new inequalities for the classical numerical radius.

abstract
Berezin numbers and their generalizations play a significant role in operator theory, particularly in the study of inequalities in Hilbert spaces. They also find applications in functional analysis, non-commutative geometry, and quantum physics. In this paper, we investigate several improvements and extensions of Berezin-type inequalities for bounded linear operators. We introduce generalized Euclidean Berezin numbers via convex functions and establish sharper bounds for classical inequalities. Furthermore, we extend weighted Davis–Wielandt inequalities and Furuta-type inequalities to the setting of semi-Hilbert spaces. These results not only enrich the theoretical framework of operator inequalities but also provide potential tools for further developments in mathematical physics and non-commutative analysis.

The quantum ground operator underlies every action of a field and that preserves the energy state of a system, maintaining the law of conservation of energy of the dynamical system given by its corresponding Lagrangian.and giving it a direction in space-time. Said operator will be a fundamental part in the system transformations in field theory and to define the field intentionality. In a quantum context, the analogy of placing a slab on a floor to stabilize it can be compared to stabilizing an energy state in a Hamiltonian potential or manifold. In both cases, a stable, lowest-energy configuration is sought. In the quantum context, this refers to an eigenstate of the Hamiltonian, which represents a stationary state of the system.

Abstract:
This paper reviews some of the results obtained by the authors in their recent work on matrix inequalities involving sectorial matrices and a matrix monotone function defined by the harmonic mean integral. The study focuses on the important concept of matrix inequalities that relate to the Schur complement of sectorial matrices with respect to a matrix monotone function.
if $A$ and $B$ are two matrices such that $ W(A) , W(B) \subseteq \mathcal{S}_{\alpha} $ for some $\alpha \in [0,\frac{\pi}{2} )$ and $\sigma_{f} $ is a matrix mean, then
\begin{align}
\Re(S(A) \sigma_{f} S(B)) \leq (\sec \alpha)^{4} \Re\textit{S}(A\sigma_{f} B).
\end{align}
Moreover, it is obtained
\begin{align}
\Re (f(S(A))) \leq (\sec \alpha)^{4} \Re S(f(A)),
\end{align}
where $f$ is a non-negative matrix monotone function with $f(1)=1$ and $A$ is a matrix such that $ W(A) \subseteq \mathcal{S}_{\alpha} $ for some $\alpha \in [0,\frac{\pi}{2} )$.

FURTHER GENERALIZED NUMERICAL RADIUS
INEQUALITIES FOR HILBERT SPACE OPERATORS

Vuk Stojiljković

$^1$ Mathematical Institute of Serbian Academy of Sciences and Arts,
Kneza Mihaila 36, Belgrade 11000, Serbia
[email protected]

We introduce a new generalized numerical radius, $w_{h,g}^{Re}(A)$, which is defined based on the generalized real and imaginary parts of an operator, as defined by Kittaneh and Stojiljković in a recent paper. We will demonstrate that this new quantity, $w_{h,g}^{Re}(A)$, is a norm on the $C^*$-algebra of bounded linear operators, $B(H)$, and that it is equivalent to the standard operator norm, $||\cdot||$. A key strength of this new concept is its generality; it encompasses and refines existing definitions and inequalities, including those previously established by Sheikhhosseini et al. and Kittaneh. Furthermore, we will explore various new inequalities, including those involving powers of operators and operator matrices, providing extensions and refinements to previous results in the field. The adaptability of $w_{h,g}^{Re}(A)$ through the functions $h$ and $g$ suggests promising applications and significant contributions to the ongoing refinement and extension of operator inequalities in functional analysis. Results in this presentation are based on the recent paper submitted by Kittaneh and Stojiljković.

In this article, we study Bohr-type inequalities involving a parameter or convex combinations for $K$-quasiconformal, sense-preserving harmonic mappings in $\mathbb{D}$, where the analytic part is subordinate to a convex function. Moreover, we establish similar inequalities when the subordinating function is chosen from the class of concave univalent functions with pole $p$, as well as from the family of concave univalent functions with opening angle $\pi\alpha$. The results generalize several existing results.

Inspired by state space theory in $C^*$-algebras, we study state spaces in Banach spaces--the family of norm-one support functionals for the unit ball at a given unit vector. This generalizes the state space of a unital $C^*$-algebra because it is the family of norm-one support functionals for the unit ball at the multiplicative unit. The state space of $X$ associated with a unit vector $x\in X$ is defined by $S_{x}=\{x^*\in X^*:\|x^*\|=1 \,\& \,x^*(x)=1\}$, where $X^*$ is the dual space of $X$. It is clear that $S_x$ is non-empty and weak$^*$-compact in $X^*$. We study the compactness of $S_x$ with respect to the norm and weak topologies on $X^*$. In this talk, we mainly discuss the norm compactness and weak compactness of the state space in the space of Bochner integrable functions and $c_{0}$, $\ell_1$-direct sums of Banach spaces.