Recent content by julydecember

  1. J

    Proving Convexity of Numerical Range for Normal T

    Homework Statement If T is a linear transformation on the finite-dimensional inner product space over complex numbers and is normal, then prove that the numerical range of T is convex. Homework Equations The Attempt at a Solution If we assume a and b are in the numerical range of...
  2. J

    Kernel of the adjoint of a linear operator

    O.K., thank you.
  3. J

    Kernel of the adjoint of a linear operator

    I think I found out how to do this. Are my steps right? Let x be an element in the kernel of T. Since TT*x = T*Tx = T*0=0, (TT*x,x) = 0 where (,) indicates the inner product. Then, (TT*x,x) = (T*x,T*x) = ||T*x||^2 = 0 so T*x = 0 and x is in the kernel of T*. The other way can be done similarly...
  4. J

    Kernel of the adjoint of a linear operator

    Homework Statement Let V be an inner product space and T:V->V a linear operator. Prove that if T is normal, then T and T* have the same kernel (T* is the adjoint of T). Homework Equations The Attempt at a Solution Let us assume x is in the kernel of T. Then, TT*x =T*Tx = T*0= 0...