On the convergence of some spectral characteristics of the converging operator sequences

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 \(\begin{aligned} \qquad \qquad \qquad \left| c(A)-c(B)\right| \le \omega (A+ e^{i\theta } B) \,\,\, \hbox { for any}\ \theta \in \mathbb {R}, \end{aligned}\) $$\begin{array}{c}\hfill \left|c,(,A,),-,c,(,B,)\right|\le \omega (A+e^{i\theta }B)\text{for any}\theta \in \mathbb{R},\end{array}$$ where \(c(\cdot )\) $c(·)$ and \(\omega (\cdot )\) $\omega (·)$ 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.