Weighted numerical radius inequalities for operator and operator matrices

In this paper, we establish some numerical radius inequalities for bounded linear operator defined on a complex Hilbert space. As a natural application, the existence of the all polynomial zeros is identified in a specific small disk. Moreover, we provide a refinement of an earlier numerical radius inequality due to Herzallah et al. [Numerical radius inequalities for certain operator matrices. Integr Equ Oper Theory. 2011;71:129–147] and a generalization of Shebrawi's inequality [Numerical radius inequalities for certain operator matrices II. Linear Algebra Appl. 2017;523:1–12].

The aim of this article is to prove several new numerical radius inequalities for n × n operator matrices on a Hilbert space. Let be complex Hilbert spaces, and let be an n × n operator matrix with . Among other inequalities, it shown thatwhere and denote the numerical radius and the usual operator norm, respectively.

We present some inequalities related to the Hilbert-Schmidt numerical radius of 2 x 2 operator matrices. More precisely, we present a formula for the Hilbert-Schmidt numerical radius of an operator as follows: w2(T) = sup ?2+?2=1 ||?A + ?B||2, where T = A + iB is the Cartesian decomposition of T ? HS(H).

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.

We introduce the new concepts of pseudo numerical range for operator functions and families of sesquilinear forms as well as the pseudo block numerical range for n!times ! n operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order 0 and not even <!1. As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping.

We consider a $2\\times2$ operator matrix $A$ (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on ${\\rm d}$-dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range $W(A)$ of this operator in detail by terms of its matrix entries for all dimensions of the torus ${\\bf T}^{\\rm d}$. Moreover, we study the cases when the set $W(A)$ is closed and give necessary and sufficient conditions under which the spectrum of $A$ coincides with its numerical range.

In this paper, new numerical radius inequalities for 2 2 operator matrices are proved. These numerical radius inequalities refine the existing upper bounds.

Numerical radius and Berezin number inequality

In this paper, using a refinement of the classical Young inequality, we present some new upper weighted bounds for the numerical radius of $2\times 2$ block matrices, with entries are bounded operators.

This paper deals with the so-called A-numerical radius associated with a positive (semi-definite) bounded linear operator A acting on a complex Hilbert space H. Several new inequalities involving this concept are established. In particular, we prove several estimates for 2×2 operator matrices whose entries are A-bounded operators. Some of the obtained results cover and extend well-known recent results due to Bani-Domi and Kittaneh. In addition, several improvements of the generalized Kittaneh estimates are obtained. The inequalities given by Feki in his work represent a generalization of the inequalities given by Kittaneh. Some refinements of the inequalities due to Feki are also presented.

We generalize several inequalities involving powers of the numerical radius for offdiagonal part of 2 2 operator matrices of the form T = 0 B C 0 , where B,C are two operators.

In this article, we employ certain properties of the transform $C_{M,m}(A)=(MI-A^*)(A-mI)$ to obtain new inequalities for the bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$. In particular, we obtain new relations among $|A|,|A^*|,|\mathfrak{R}A|$ and $|\mathfrak{I}A|$. Further numerical radius inequalities that extend some known inequalities will be presented too.

Delay-dependent stability of descriptor neutral systems with mixed delays is investigated in this paper. Based on descriptor system approach, some new delay-dependent stability and robust stability criteria are established in terms of a operator and linear matrix inequalities(LMIs). Lyapunov-Krasovskii functional and Leibniz-Newton formula are applied to find the stability results.

This study focuses on the mean square couple-group consensus of a heterogeneous multi-agent system with time delay and Markov switching. Based on heterogeneity, time delay, competitive-cooperative relations, and stochastic switching, new protocols have been developed to achieve the mean square couple-group consensus of this system. Sufficient conditions have been achieved for the mean square couple-group consensus of this system by using the properties of stochastic indecomposable and aperiodic matrices, stochastic delta operator, graph theory, and linear matrix inequality techniques. These findings demonstrate that some strict demands for the topology, such as strong connectivity or a spanning tree, can be relaxed. The reliability of our findings has been evaluated by several numerical experiments.

We present an expression for the generalized numerical radius associated with a norm on the algebra of bounded linear operators on a Hilbert space and then apply it to obtain upper and lower bounds for the generalized numerical radius. We also establish some generalized numerical radius inequalities involving the product of two operators. Applications of our inequalities are also provided.