I've been thinking about this question since I submitted it and I think I might know the answer now, but I would appreciate an expert opinion on my answer.

I think I am still right about the redshift turning into a blue shift as the infalling observer approaches the event horizon. I don't know for sure if the photons he sees as he crosses the event horizon will be back to no shift at all or if there would still be a net redshift or even a blueshift - does anyone know?

But I now think that the surface of the star will look dimmer and dimmer as the infalling observer approaches the event horizon. Also the observer will only be able to see a smaller and smaller patch of the surface of the star directly below him in the radial direction as he approaches the event horizon.

The reason for this is that as the original star approached the event horizon, only photons emitted in an increasingly narrow cone along the radial direction will be able to avoid falling into the black hole. At the event horizon, only photons emitted exactly along the radial direction will remain frozen at the horizon waiting for the infalling observer to see.

Once the observer enters past the event horizon he will not see anything at all since the forward light cone points inward in a "spatial" direction towards the singularity.

Finally, these consideration affect what the observer at infinity sees - as the star surface approaches the event horizon, not only will the photons become more and more redshifted, but in addition, the patch of the star that the observer at infinity can see will be a smaller and smaller disk that will shrink towards a point at infinite time.

Is this right?

The redshift turns into a blue shift as the infalling observer approaches the event horizon. I don't know for sure if the photons he sees as he crosses the event horizon will be back to no shift at all or if there would still be a net redshift or even a blueshift - does anyone know?

But the surface of the star will look dimmer and dimmer as the infalling observer approaches the event horizon. Also the observer will only be able to see a smaller and smaller patch of the surface of the star directly below him in the radial direction as he approaches the event horizon.

The reason for this is that as the original star approached the event horizon, only photons emitted in an increasingly narrow cone along the radial direction will be able to avoid falling into the black hole. At the event horizon, only photons emitted exactly along the radial direction will remain frozen at the horizon waiting for the infalling observer to see.

Once the observer enters past the event horizon he will not see anything at all since the forward light cone points inward in a "spatial" direction towards the singularity.

Finally, these consideration affect what the observer at infinity sees - as the star surface approaches the event horizon, not only will the photons become more and more redshifted, but in addition, the patch of the star that the observer at infinity can see will be a smaller and smaller disk that will shrink towards a point at infinite time.