Hidden/Floating Hardware for Solid Wood Alcove Shelves?
Edit:  I didn't read the requirement that it be supported on the ends.

The easiest way to support it from the ends would be with z-clips
 

But they are 2 1/8 wide which is likely too wide.

Tiberhan clips appear to be smaller.  You may be able to recess it into the end of the board to reduce the gap between it and the wall. https://www.amazon.com/Bracket-Tiberham-Interlocking-Stainless-Hardware/dp/B07K6HVYP5  No weight limit is listed.  The failure mode I think would be the screw in the drywall side bending.  A common #8 screw typically has a sheer strength rated at 75 pounds.  With 2 brackets on each end, that gives 4 screws at each end, allowing for 600 lbs.



This, however still leaves a bit of a gap at the wall end.

How about this for an idea:  For a 9.5 inch shelf (2x10) you put a 9" long section of 1" angle iron onto the wall.  The end of your shelf has a pair of blind dados that would accept the angle iron and be flush. Shelf slides in from the front, but is supported on the ends.

Original answer for free floating shelf against a solid wall.

This is actually fairly easy.


Find your studs.  find them for the entire length of the stud -- don't assume that studs are vertical.
Use a laser level and mark your holes.  Also mark a line 1/2 the shelf thickness below or above your line of holes.  
Drill an initial hole to match elements of your set of dowel centres.  https://www.leevalley.com/en-ca/shop/tools/hand-tools/marking-and-measuring/marking-tools/44995-dowel-and-tenon-centers.  I suggest using a jig to make sure that these are perpendicular to the wall.
Place the dowel centres plug ends in the holes.
This step is easier with 2 people:  Get your board lined up with the line, and pressed against the dowel centres.  
Verify that the board is lined up with the line.
Use a mallet and tap the shelf on the edge opposite each dowel centre.  Recheck that you are on the line after each tap.
Get a supply of 1/2" steel bar stock.  Verify that it's diameter is consistent. Cold rolled is more accurate.  You can also use 1/2" diameter bolts
Redrill your wall holes to match the steel bars.  If using bolts initially drill about 1/64" under the nominal bolt size.  This will allow the bolt threads to bite into the stud.
Drill your shelves at the dowel centre mark to match the amount of exposed shaft.  I would use nominal 7 inch bars so that you have 3 inches in the stud, 1/2" in the drywall and 3.5 inches in the shelf.  Use a drill press to get perpendicular holes.
If you want the shelves to be portable, put the threaded end in the shelf, and glue in place with epoxy.  
Put the shelf into the matching holes.




Risks:  wires are often run through studs at the centre, 1.5" from the stud's surface.  If you are unlucky you may drill through a wire.  For this reason, I would not put a shelf within 1 foot vertically of an outlet, or within 2 studs of a light switch.

You may find the end of a screw for the room on the other side of the wall's drywall, or the outside wall sheathing.  This will pop the screw or nail and will require some inconvenient repair work.



Cool option:  Use a longer rod, and paint it a lighter version of the wall colour.  (Diluting the wall paint with 1-2 parts white works well)  Now the shelf is really floating:  The back edge of the shelf is 1 to 2 inches away from the wall.  The light painted bars are remarkably hard to see.  This works best when the shelf is NOT in an alcove, but both ends are just ending in the middle of the wall.

Option:  I find that 1.5" thick shelves look clunky.  If you have access to a bandsaw with re-saw capability, you can make shelving out of 2x12, and using a jig to hold it at an angle, slice the board in half along the width, giving  you 2 triangular sections, each 1.5" thick on one edge to essentially zero on the other.