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Math Help - Surjective Operator on Hilbert Space

  1. #1
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    Surjective Operator on Hilbert Space

    I have an interesting problem that I've been stewing over today, but I can't come up with any examples:

    Find a surjective operator A:H -> H such that ker(A) is non-trivial, where H can be any example of a Hilbert space.

    Thanks in advance for any help.
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  2. #2
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    Quote Originally Posted by joeyjoejoe View Post
    I have an interesting problem that I've been stewing over today, but I can't come up with any examples:

    Find a surjective operator A:H -> H such that ker(A) is non-trivial, where H can be any example of a Hilbert space.
    The easiest example is the backward shift operator S on l^2(\mathbb{N}), defined by S(x_1,x_2,x_3,\ldots) = (x_2,x_3,x_4\ldots). That is fairly obviously surjective, and the vector (1,0,0,...) is in the kernel.
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