Some time ago, my trusty HP 8595E spectrum analyzer (8590 E Series) would not come on. It turned out that the power supply was faulty. A quick search on the internet revealed that this is a recurring thing, many people are looking for a fix or a new PSU. Even though HP (or Agilent, if you will) documents the instrument thoroughly the A8 Power Supply (part # 5062-8229) is listed as "replace or exchange when faulty" only. Of course, spares are exhausted by now. So I decided to open the instrument and look inside the PSU. I saw a blown up tantalium capacitor:
Easy peasy, I thought. Not so. I really had to delve into the complex thing. What a nightmare. Eventually (failed to document properly, sorry 'bout that) I found that the NE5560 startup resistors were high in value, thus depriving this (also) complex chip of its startup current. Replacing the two resistors with new SFR25 metal film ones brought my beloved instrument back to life. Phew!
Note that the fault was not on the sub-board with the controller (photo off the net, apologies for not giving credit):
Of course, YMMV but it's worth a shot. Also, your PSU may be a different design. It will be found in all the variations of this type, HP 8591E, 8593E, 8594E and so on. Oh, and the tantalium caps were replaced with lo-ESR aluminium types. After all, without them the analyzer worked just fine anyway. And finally, unplug the instrument when it's not in use. Because it is still powered on but in stand by while switched off it could fail again. Maybe with another failure mode.