Orthogonality of filter impulse response to its even shift

I meet this problem but still do not know how to solve it. Could you guy give me some guides?

Upsampling by 2 ($$U_2$$) followed by filtering by $$g$$, with operator $$G$$

And given: $$_n = \delta_k$$ (Filters with impulse responses orthogonal to their even shifts)

Prove that: $$I = U_2^*G^*GU_2$$

• The notation you're using is unfamiliar. What is $\left < g_n,\,g_{n-2k} \right >_n = \delta_k$ supposed to mean? If you don't know, there's your problem, and it should be in your book or the lecture notes. – TimWescott Mar 2 '20 at 1:09
• @TimWescott I'd have said that these brackets signify the signal vector space inner product between an infinite sequence $g[n]$ and it's $2k$-shifted variant $g[n-2k]$. And that's always 0, unless $k=0$. – Marcus Müller Mar 2 '20 at 9:01