If D2, which is nearly stainless and doesn't oxidize easily, those marks look a lot like etching from something acidic or caustic. Such etches can take on a 'milky' appearance on stainless steel. If so, it might take some abrasive work with fine sandpaper (wet/dry type) to remove them. A satin finish is usually in the 320-800 grit range, for wet/dry sandpaper. Metal polishes like Flitz or Simichrome, or Mother's Mag wheel polish, might (or might not) work entirely, depending on how deep the etch goes. If they don't work on their own, then the sandpaper treatment would likely be the best fix. Try Flitz or one of these other aluminum oxide-based metal polishes which are made to work on steel first. Toothpaste likely won't touch that, if it's etched on D2; nor will a silverware polish, which likely isn't aggressive enough to polish D2 steel.
David