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By definition, $$\langle T(t)x,E_0x\rangle_H=\left\|E_0x\right\|_H^2+\underbrace{\int_0^\infty e^{-t\lambda}\:{\rm d}\underbrace{\langle E_\lambda x,E_0x\rangle_H}_{=\:\left\|E_0x\right\|_H^2}}_{=\:0}\tag5$$ and hence $$\left\|T(t)x-E_0x\right\|_H^2=\left\|T(t)x\right\|_H^2+\left\|E_0x\right\|_H^2\xrightarrow{t\to\infty}0\tag6$$$$\left\|T(t)x-E_0x\right\|_H^2=\left\|T(t)x\right\|_H^2-\left\|E_0x\right\|_H^2\xrightarrow{t\to\infty}0\tag6$$ for all $x\in H$.

By definition, $$\langle T(t)x,E_0x\rangle_H=\left\|E_0x\right\|_H^2+\underbrace{\int_0^\infty e^{-t\lambda}\:{\rm d}\underbrace{\langle E_\lambda x,E_0x\rangle_H}_{=\:\left\|E_0x\right\|_H^2}}_{=\:0}\tag5$$ and hence $$\left\|T(t)x-E_0x\right\|_H^2=\left\|T(t)x\right\|_H^2+\left\|E_0x\right\|_H^2\xrightarrow{t\to\infty}0\tag6$$ for all $x\in H$.

By definition, $$\langle T(t)x,E_0x\rangle_H=\left\|E_0x\right\|_H^2+\underbrace{\int_0^\infty e^{-t\lambda}\:{\rm d}\underbrace{\langle E_\lambda x,E_0x\rangle_H}_{=\:\left\|E_0x\right\|_H^2}}_{=\:0}\tag5$$ and hence $$\left\|T(t)x-E_0x\right\|_H^2=\left\|T(t)x\right\|_H^2-\left\|E_0x\right\|_H^2\xrightarrow{t\to\infty}0\tag6$$ for all $x\in H$.

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0xbadf00d
0xbadf00d
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By definition, $$\langle T(t)x,E_0x\rangle_H=\left\|E_0x\right\|_H^2+\underbrace{\int_0^\infty e^{-t\lambda}\:{\rm d}\underbrace{\langle E_\lambda x,E_0x\rangle_H}_{=\:\left\|E_0x\right\|_H^2}}_{=\:0}\tag5$$ and hence $$\left\|T(t)x-E_0x\right\|_H^2=\left\|T(t)x\right\|_H^2+\left\|E_0x\right\|_H^2\xrightarrow{t\to\infty}0\tag6$$ for all $x\in H$.