0 for all T (s ) .

3, there exists uniquely u E H such that f(x) (Tx , y ) = (x , u ) for all x = adjoint operator of T , by (Tx , y ) Theorem 1. = (x , u ) = H. Hence, we may define T* , the E (x , T* y ) for x , Y E H. Let T be an operator on a Hilbert space H. Then T* is also an operat or on H, and the following properties hold: (i) I I T' I I (ii) (Tl + T2 )' (iii) (aT)" (iv) (T* )" = T. (v) (ST)* = T*S* . Proof. If Yb Y2 E = II T I I · = = ar" H and a, /3 (x , T* (aYl + /3Y2» for any a E C. Yl + /3Y2 ) E H, a(Tx , Yl ) + /3(Tx , Y2 ) = ( x, aT*Yl + /3T · Y2 ) .