Question

Problem 5. (10 points total) For a linear operator l' on a complex inner product space define (a) (2 points) Show that T+ and T- are self-adjoint and T=T+ +iT-. (b) (3 points) ow that the representation in part (a) is unique. (c) (5 points) Show that T is normal if and only if T+T TT

Answer 1

Given: A linear operator T on a complex inner product space define T + = 1/2 (T + T*) - 1 T- = 1/2s (T - T*) - 2 A bounded linear operator T: H rightarrow H on a Hilbert space H is said to be self adjoint or Hermitian if T* = T normal if TT* = T* T (T +)^* = (1/2(T + T*))^* = 1/2 (T + T^*)^* = 1/2 (T^* + T^**) = 1/2 (T + T^*) = T + Rightarrow (T +) = T + (T-)^* = (1/2i (T - T^*))^* = 1/2i^- (T - T^*)^* = 1/- 2i (T^* - T^**) = -1/2i (T^* - T) = 1/2i (T - T^*) = T Rightarrow (T-)^* = T- Hence T + and T are self - adjoint \r\n from 1 and from 2 T + T^* = 2T + - 3 T - T^* = 2i T- - 4 Adding 3 and 4 2T = 2T + + 2i T- Rightarrow T = T + + i T- proved let T = T + + i T = S + + i S-