Ps the carpet is dark grey / black

Maybe some large colourful artwork on the walls, and maybe paint the inside of your door a darker colour.

I would also paint the trim by the carpet a white. The trim draws your eye to it.

I would paint the walls the color of the sky!

More info about the LED lighting under the stairs, please. I have this problem too. Thanks!

Agreed, replacing the carpet with wood with make a huge impact to brighten up the space.

Yes that is a very dark carpet! I like the idea about lighting at the floor level with LEDs that sounds like a great idea. I would go for a warmer tan kinda colour on the walls. And lots of bright modern artwork on the walls. The great thing about dark colours is you can balance them with strong bright colours. Art is a great way to make a hallway inviting. Try looking out for bright vintage posters and comic book art.
The light overhead seems quite bright and harsh, I think it would create shadows. i would change the light bulb on the light to make it a warmer, soft yellow glow. And I would try lighting the treads as suggested. Or perhaps painting them bright colours? Does the carpet run down the stair as well? Could you add a pop of colour to brighten it? A deep turquoise could look good.
Your house looks art deco style, I would look to that era for bold geometric patterns and colours to match. Look up Bauhaus style for inspiration. Its a great era for a mix of old materials and modern lines.
I found this awesome picture which showed backlight artglass. I think that would certainly add colour and light into the hallway!
You could also try gold frames around the art instead of black, it brightens the room.
I even found one photo where they just embraced the dark look and trimmed in a crisp white and used lovely lights!
I guess it just depends what you like!

Art Deco

http://www.mokumtextiles.com/metropolis---catherine-martin-by-mokum