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Let $ \mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $ F$ . For $ M\in \mathcal{B}(F)^+$ , we consider $ $ \mathcal{B}_M(F)=\left\{S\in \mathcal{B}(F):\,\,\exists c>0 ;\;\|Sx\|_M \leq c \|x\|_M ,\;\forall x \in \overline{\text{Im}(M)}\right\}.$ $ with $ \|x\|_M:=\|M^{1/2}x\|,\;\forall x \in F$ . If $ S\in \mathcal{B}_M(F)$ , the $ M$ -semi-normRead more