I can't tell from the photos, but it's either bugs, fungus or fertilizer.
Bugs are not common on Sylvester Date Palms (that's what it looks like) and it's not been so wet for funger (though it could be - copper fungicide from Home Depot works great).
Last, you need to fertilize 3 or 4 times a year. Lesco 8-2-12 or Lutz spikes, both have mangesium and manganese, which these trees (and all palms) need.
Cut off any dead, it won't grow back. Best wishes.
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