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Some spectral properties and convergence of the $ (A,q)$-numerical radius and $ (A,q)$-Crawford number

Pembe Ipek Al, Zameddin I. Ismailov, Fuad Kittaneh, Satyajit Sahoo

Abstract

In this study, some estimates are given for the $ (A,q)$-numerical radius and $ (A,q)$-Crawford number via the $ A$-numerical radius and $ A$-Crawford number for the $ A $-bounded linear operators in any complex semi-Hilbert space, respectively. Then, some evolutions are studied for the tensor product of two operators. Lastly, some convergence properties of the $ (A,q)$-numerical radius and $ (A,q)$-Crawford number, via the $ A$-uniform convergence of operator sequences, are investigated. We also considered several examples to illustrate our results. Finally, a few applications of some operator functions classes are also given.

Some spectral properties and convergence of the $ (A,q)$-numerical radius and $ (A,q)$-Crawford number

Abstract

In this study, some estimates are given for the -numerical radius and -Crawford number via the -numerical radius and -Crawford number for the -bounded linear operators in any complex semi-Hilbert space, respectively. Then, some evolutions are studied for the tensor product of two operators. Lastly, some convergence properties of the -numerical radius and -Crawford number, via the -uniform convergence of operator sequences, are investigated. We also considered several examples to illustrate our results. Finally, a few applications of some operator functions classes are also given.

Paper Structure

This paper contains 4 sections, 10 theorems, 112 equations, 4 figures.

Table of Contents

  1. 1. Introduction
  2. 2. Some properties of the $(A,q)$-numerical radius and $(A,q)$-Crawford number
  3. 3. Some convergence properties of the $(A,q)$-numerical radius and $(A,q)$-Crawford number via operator sequences
  4. 4. Some applications

Key Result

Theorem 1

For the operator $T\in L^{A}(H)$ and $q,\lambda,\mu\in \mathbb{C}, \ 0<\vert q \vert\leq 1,$ the following are true (1) $\omega_{A,q}(T)\leq \Vert T \Vert_{A}.$ (2) If $\alpha \in \mathbb{C}, \ \vert \alpha \vert =1,$ then $\omega_{A,q}(\alpha T)=\omega_{A,\alpha q}(T).$ (3) If $\alpha \in \mathbb{C (7) $\omega_{A}(T)\leq \omega_{A, q}(T)+ \sqrt{2(1-Req)}\omega_{A,\frac{1-q}{\sqrt{2(1-Req)}}}(T),

Figures (4)

  • Figure 1: Comparision of $\omega_q(T)$ with the upper and lower bounds \ref{['Eqn_2']} for Example \ref{['Ex_1']}.
  • Figure 2: Comparision of $\omega_q(T)$ with the upper bound \ref{['equ2']} for Example \ref{['Ex_2']}.
  • Figure 3: Comparision of $\omega_q(T)$ with the upper bound \ref{['equ2']} for Example \ref{['Ex_3']}.
  • Figure 4: Comparision of $\omega_q(T)$ with the upper bound \ref{['equ2']} for Example \ref{['Ex_4']}.

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Remark 1
  • Example 1
  • Theorem 3
  • proof
  • ...and 15 more