Sure...we use certain power tools and hand tools to get the job done...from reciprocating saws with long and sharp blades, to simple hand operated Japanese-type pull saws. Reciprocating saws can take some skill and practice to make these cuts accurate and clean. Hand operated Japanese-type pull saws offer more control for the novice, but take longer to make the cut. But both work here.

Make sure you have a few terribly sharp chisels to shave and cut away any rough edges and to clean out the cut.

However...why do you want to provide for a transition piece the way you envision? A suitable transition piece in the correct depth and pitch ought to do the job...with no need to cut into the existing wood flooring. the height of the transition would be the difference in height between your tile and the wood. It is tapered from the depth at the butt edge to an almost feather edge where it meets the wood. Any custom wood mill ought to be able to fabricate what you want.

Thanks guys. You are correct I could install a transition for the current situation but the angle of the change would be severe and provide a tripping hazard. Pushing it into the kitchen a few more inches would make it less noticeable.

I know of Japanese pull saws but both sides of the passageway end at cabinets and would be difficult to end the cut cleanly. Would that still work? Can you give me some hints on how to start the cut and a link so that I can see the best sort of saw?

If you have access to a plunge track saw and a track of appropriate length, you could use that to make your first cut. You can use a circular saw to do that, but you need to provide a 'fence' to allow for a straight cut. Think about it for awhile, look at what you want to do and the solution will come.

Once you have the first cut done, then use the pull saw or another hand-operated sharp saw to cut from the first cut to the cabinet. Place a thin piece of metal or plastic laminate up against the vertical surface of the cabinet to avoid marring that with the tip of the saw.

Sharp wood chisels are used to dress the cut, chip out areas where the saw blade can't reach and to clean out the recess you've made.

I don't have any links. Search youtube using possible descriptions of what you want to do and I am sure there will be stuff there to give you some tips.

Or, hire a competent wood flooring specialist.

those plunge track saws are pricey! Maybe I can make do with my circular saw and a sharp blade. I get the feeling that if I get the cut going in the middle of the floor, that I can extend it eventually to the cabinet edges, working slowly and carefully.