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3 1 $begingroup$ Let $(E,langlecdot;,;cdotrangle)$ be a complex Hilbert space and $mathcal{L}(E)$ be the algebra of all operators on $E$ . Assume that $Tin mathcal{L}(E)$ and satisfy the following property (P): $$langle x,yrangle=0Longrightarrow langle Tx,Tyrangle=0,$$ for all $x,yin E$ . I want to prove under the property $(P)$ that there exists $cgeq0$ such that $$|Tx|=c|x|,$$ for all $xin E$ . Note that, if there exist $x, yin Esetminus{0}$ such that $|Tx|=alpha|x|$ and $|Ty|=beta|y|$ for some $0leqalpha <beta$ , it can be seen that $langle x,yrangleneq0$ . functional-analysis operator-theory hilbert-spaces share | cite | improve this question