(adsbygoogle = window.adsbygoogle || []).push({}); A functional analysis' problem

I hope this is the right place to submit this post.

1. The problem statement, all variables and given/known data

Let [itex]A[/itex] be a symmetric operator, [itex]A\supseteq B[/itex] and [itex]\mathcal{R}_{A+\imath I}=\mathcal{R}_{B+\imath I}[/itex] (where [itex]\mathcal{R}[/itex] means the range of the operator). Show that [itex]A=B[/itex].

2. The attempt at a solution

If [itex]A[/itex] is symmetric, then [itex]A\subseteq A^*[/itex], where [itex]A^*[/itex] is the adjoint of [itex]A[/itex], and from [itex]A\supseteq B[/itex] one can deduce that also [itex]B[/itex] is symmetric. The definition domains of [itex]A[/itex] and [itex]B[/itex] are dense in [itex]\mathcal{H}[/itex] ([itex]\mathcal{H}[/itex] an Hilbert space), so [itex]\mathcal{N}_{(A+\imath I)^*}=(\mathcal{R}_{A+\imath I})^\perp[/itex], where now [itex]\mathcal{N}[/itex] is the operator's kernel. An idea to complete the exercise should be showing that [itex]A^*\supseteq B^*[/itex], using the identity [itex]\mathcal{N}_{(A+\imath I)^*}=\mathcal{N}_{(B+\imath I)^*}[/itex] and the previous hypothesis. However, I can't understand how this could be usefull.

Thanks a lot for your help. JM

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# A functional analysis problem

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