I have a question about how to show these equalities, $$|||T|||=|||T^{*}|||=||T||$$ and $$|||T||T^{*}|||=||T^{2}||$$ where $$|T|=(T^{*}T)^{\frac{1}{2}},|T^{*}|=(TT^{*})^{\frac{1}{2}}.$$ First guess is to use the definition $$\langle (T^{*}T)^{\frac{1}{2}}x,x\rangle = \langle x,(T^{*}T)^{\frac{1}{2}}x\rangle$$ But I dont seem to get how to connect it with $$(TT^{*})^{\frac{1}{2}}$$
Thank you so much for helping!