Inverse of the adjoint of the shift operator

[Likemath2014]

Hi there,

Let S denote the shift operator on the Hardy space on the unit disc H^2, that is (Sf)(z)=zf(z).

My question is to show the following identity

(1-\lambda S^*)^{-1}S^*f (z)=\frac{f(z)-f(\lambda)}{z-\lambda},

where \lambda,z\in\mathbb{D}

Thanks in advance

 

[micromass]

First of all, can you figure out what $S^*$ does exactly? This can be made easy if you can figure out an orthonormal basis of $H^2$.

 

[Likemath2014]

Yes, S^*=\frac{f(z)-f(0)}{z}. But my problem is with the term (1-\lambda S^*)^{-1}.

 

[micromass]

Just put it on the other side. So you need to prove

S^*f(z) = (1-\lambda S^*)\frac{f(z)-f(\lambda)}{z-\lambda}