# Thread: Surjective Operator on Hilbert Space

1. ## 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.

2. Originally Posted by joeyjoejoe
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.