Iyad's Page of Operators

Let H be a Hilbert space and let A: H ---> H be an operator on H and a any nonzero real number. A is said to be:

  1. Self-adjoint if <Af,g>=<f,Ag>, for all f in H and all g in H.
  2. Positive if <Af,f>>=0, for all f in H.
  3. Normal if A*A=AA*, for all f in H and all g in H.
  4. Isometry if ||Af||=||f||, for all f in H.
  5. Unitary if U-1=U*.
  6. Coercive (or bounded away from zero) if there exists a positive constant C such that ||Af|| >= C ||f||, for every nonzero element f of H.
  7. Bounded if ||A|| is finite. The norm of A is defined to be:
    ||A||={sup ||Af||: f in H, ||f||=1}.

  8. The translation operator is defined as follows:
    Taf(x)=f(x-a).

  9. The modulation operator is defined as follows:
    Eaf(x)=e2 pi i a xf(x).

  10. The dilation operator is defined as follows:
    Daf(x)=(1/sqrt(|a|))f(x/a).

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