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Math Help - Example of operators

  1. #1
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    Example of operators

    Do you know the example of next operators:
    B:H_1\to H_2 is a closed operator, A:H_2\to H_3 is a bounded linear operator such that
    AB is not a closable operator?
    H_1,H_2,H_3 are Hilbert spaces.
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  2. #2
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    Quote Originally Posted by karkusha View Post
    Do you know the example of next operators:
    B:H_1\to H_2 is a closed operator, A:H_2\to H_3 is a bounded linear operator such that
    AB is not a closable operator?
    H_1,H_2,H_3 are Hilbert spaces.
    Let \{e_n\}_{n=1}^\infty be an orthonormal basis for the Hilbert space H. Define A, B on H by Ae_n = \frac1ne_1 and Be_n=n^2e_n (for all n). Then B is closable (it is densely defined and contained in its adjoint), A is bounded (use Cauchy–Schwarz to show that), but AB is not closable (if x_n = \frac1ne_n then x_n\to0 but ABx_n\to e_1).
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  3. #3
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    Thank you for the reply, but Is A linear operator?
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  4. #4
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    Quote Originally Posted by karkusha View Post
    Is A linear operator?
    I should have said that you need to extend both A and B by linearity to the space of all finite linear combinations of the basis vectors, and then take their closures. So if x = \sum\lambda_ne_n (only finitely many \lambdas nonzero) then Ax = \sum\frac{\lambda_n}ne_1, and Bx = \sum n^2\lambda_ne_n. That way, A and B are both linear.
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