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
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[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.
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[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.

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.

Identifying the isometries on a normed linear space is one of the fundamental goals in functional analysis and operator theory. The classical Blanco–Koldobsky–Turn\v{s}ek theorem characterizes the isometries as the norm-one linear operators that preserve Birkhoff–James orthogonality. In this study, we consider the local preservation of Birkhoff–James orthogonality by linear operators between normed linear spaces, both at a point and in a particular direction. We obtain a complete characterization of the same, refine earlier results, and apply these findings to the identification of isometries on polyhedral Banach spaces. In particular, we present refinements of the Blanco–Koldobsky–Turn\v{s}ek theorem for certain polyhedral Banach spaces.

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.

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.

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.

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.

By a systematic development of fundamental concepts of conformable calculus we establish conformable divergence theorem and Green's identities which we combine with some new anisotropic Picone-type identities to derive a generalized anisotropic Hardy type inequality with weights and conformable fractional differential operators. As a consequence, several Hardy type inequalities and Heisenberg Pauli-Weyl uncertainty principles are obtained.

Keywords: Fractional derivatives, Green's identities, Picone identity, Hardy inequalities, uncertainty principles

Representations induced by general positive sesquilinear maps with values in ordered Banach bimodules such as commutative and non-commutative
L1-spaces and the spaces of bounded linear operators from a
C-algebra into the dual of another C-algebra are considered. As a starting point, a generalized Cauchy-Schwarz inequality is proved for these maps and a representation of bounded positive maps from a (quasi) -algebra into such an ordered Banach bimodule is derived and some more inequalities for these maps are deduced. In particular, an extension of Paulsen’s modified Kadison-Schwarz inequality for 2-positive maps to the case of general positive maps from a unital -algebra into the space of trace-class operators on a separable Hilbert space and into the duals of von-Neumann algebras is obtained. Also, representations for completely positive maps with values in an ordered Banach bimodule and Cauchy-Schwarz inequality for infinite sums of such maps are provided. Concrete examples illustrate the results.

In this talk, we provide a complete characterization of the convexity of the Berezin range of composition operators on weighted Bergman spaces. Moreover, we show that while the origin lies in the closure of the Berezin range, it does not belong to the range itself.