generates the norm topology on $\B(\H_0)$ in this case $\square$.

Different types of operator convergence can be obtained by using the norm $\|\!\cdot\!\|_E^{G}$ induced by different operators ${G}$.

**Example.** Let $\H=\H_1\oplus\H_2$ and ${G}={G}_1\oplus {G}_2$, where ${G}_k$ is a positive densely defined operator on a separable Hilbert space $\H_k$ satisfying condition (\[H-cond\]), $k=1,2$. By using definition (\[ec-on\]) and the triangle inequality it is easy to show that $$\label{E-ineq}
  \sqrt{p\!\left[\|AP_1\|^{{G}_1}_E\right]^2+(1-p)\!\left[\|AP_2\|^{{G}_2}_E\right]^2}\leq \|A\|^{G}_E\leq \|AP_1\|^{{G}_1}_E+\|AP_2\|^{{G}_2}_E$$ for any $p\in[0,1]$, where $P_k$ is the projector on the subspace $\H_k$ and $\|AP_k\|^{{G}_k}_E$, $k=1,2$, are defined in accordance with Remark \[en-gen\].

Assume that ${G}_1$ is a discrete unbounded operator (Def.\[D-H\]) and ${G}_2$ is a bounded operator. Then it follows from (\[E-ineq\]) and Proposition \[ec-on-p2\] that $$\left\{\|\!\cdot\!\|^{G}_E{\hspace{1pt}}\textrm{-}\lim_{n\rightarrow\infty} A_n=A_0\right\}\quad \Leftrightarrow\quad \left\{{s.o.}{\hspace{1pt}}\textrm{-}\lim_{n\rightarrow\infty} A_nP_1=A_0P_1\right\} \wedge \left\{\|\!\cdot\!\|{\hspace{1pt}}\textrm{-}\lim_{n\rightarrow\infty} A_nP_2=A_0P_2\right\}$$ for a bounded sequence $\{A_n\}\subset\B(\H)$, where ${s.o.}{\hspace{1pt}}\textrm{-}\lim$ denotes the limit w.r.t the strong operator topology. So, in this case the norm $\|\!\cdot\!\|^{G}_E$ generates a “hybrid” topology on bounded subsets of $\B(\H)$ – some kind of the Cartesian product of the strong operator and the norm topologies.

Equivalent definitions and equivalent norms
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Recall that $\T_{+,1}(\H)$ denotes the positive part of the unit ball in $\T(\H)$. Denote by $\H_{\leq 1}$ the unit ball in $\H$.

\[en-def\] A) 