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The Product is densely defined?

  1. Aug 14, 2016 #1
    1. The problem statement, all variables and given/known data
    Hello,
    We know that if A and B are two unbounded densely defined operators, it does not mean that AB is also densely defined. But if A is bounded then D (AB) = D (B) ie AB is densely defined.
    Is AB densely defined if:
    1) B is bounded and A is unbounded densely defined operator.
    2) A and B are unbounded densely defined operators such that A or B is invertible of a bounded inverse.
    thank you.
    3. The attempt at a solution
    p_226qysu1.png
     
    Last edited: Aug 14, 2016
  2. jcsd
  3. Aug 14, 2016 #2

    Krylov

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    I think here you mean ##D(A B) = D(B)##?
    Yes, ##B^{-1}## has a dense range, which means that ##B^{-1}X## is dense (where ##X## is the underlying Banach space), but ##D(A)## is smaller than ##X## so it is not clear that ##B^{-1}D(A)## is dense as well.
     
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