Thanks everyone for your efforts... Problem Solved!

 Awhile back the e-mail provider connected to my problem mailbox decided to drop POP3 and use only IMAP.  I created an IMAP profile, but had so many problems trying to get IMAP to work with this provider that I finally gave up, set all e-mail from this provider to forward to another provider I use that supports POP3, and had Pegasus retrieve my e-mail from the POP3 client.  In reading over all your replies, a reference to IMAP made me realize that the persistent folder problem began about the same time as the IMAP adventure.  I checked, and sure enough I still had an IMAP profile for the original e-mail provider entered for that mailbox in Pegaus.. It turns out that it was the IMAP profile that was opening the "folders" window.  After I eliminated the profile from IMAP in that mailbox on Pegasus, the persistent window disappeared.

Thanks again for all your suggestions... and research on my behalf.  Hope you all have a great holiday season.

Pegasus current version on Win 10, non-maximized window.  When ever I open Pegasus on one of my accounts, the Pegasus desktop comes up normal except the "Folders" window is open with enormous text.  The other windows have normal size text.  I close this "folders" window then close Pegasus.  When I open Pegasus again... the "folders" window is there again.  I have  "Save the Pegasus Mail desktop state between sessions" checked in basic settings.  I tried resetting the font in the "folders" window to an appropriate size, close folders window, close Pegasus mail.  Upon again opening Pegasus mail, the "folders" window is there again, but now the font is still the correct size.  Any idea how I can close the "folders" window and keep it from opening again on the next session?

[quote user="jficklin"]When ever I open Pegasus on one of my accounts, the Pegasus desktop comes up normal except the "Folders" window is open with enormous text.  The other windows have normal size text.[/quote]

Folders window can have its Fonts adjusted. As a matter of fact it is one of those (few?) windows that can have its fonts configured. Click on the window and you will see a Folders menu item appear at Pegasus Mail menu bar. Pick the Font... option on it and choose a suitable one. Once your Folders pane is set close it to save settings (Pmail main window should not be maximized). Ctrl+L will restore it.

[quote user="jficklin"]Any idea how I can close the "folders" window and keep it from opening again on the next session?
[/quote]

As far as I know, this is not possible and I don't see why it should. If you think it would help, try Preview mode.

Pegasus Mail a folder window closed status should stick between sessions.

The main Pegasus Mail window isn't maximized so that rules out the associated issue with saving the desktop state.

A mention of "one of my accounts" was made.  I'm curious about what this means.

Could the startup of Pegasus Mail be as one identity but it's another identity when Pegasus Mail is closed? 

Perhaps the option to auto-open groups with unread mail is enabled.  I don't know whether this would cause the auto-open of the folder window but such behavior would make sense.

Brian, thanks for reading my whole post before responding. 

When I say "one of my accounts" I refer to opening Pegasus as the user of one of my e-mail accounts.  I have several e-mail accounts and have Pegasus configured for each.  The instance of Pegasus configured for the account where this is happening is configured identically (except for e-mail account connection info) to the others.  None of the others behave in this manner.

When I start Pegasus as this identity it also closes as the same identity.

 Auto open of groups is unchecked for all options.

Do you happen to know what file stores the desktop state between sessions?  Perhaps, I need to delete this file and force Pegasus to recreate it.

This problem doesn't really affect the usability of Pegasus, it's just annoying to have that extra window open and/or annoying to have to close it every time I run the program as that identity.

 Thanks for your response.

Euler, thanks for your reply.  As I mentioned in my post as was able to change the "folders" window fonts to an appropriate size and the change persisted between sessions.  It is the closing of the folders window that should, but does not persist between sessions.

Brian, or any other who knows,

 Would the file that stores desktop status be the following:

PMAIL > MAIL > (name of mailbox with problem) > FOLSTATE.PM

If so, do you know whether or not if I deleted it, Pegasus would simply recreate it at the next session...

or might Pegasus 1. crash, 2. recreate the whole folder, 3. lose all the mail stored in this mailbox, or 4. something else, like leave the folder missing this file.

 I, of course, would back up the entire mailbox folder before trying this, and should be able to restore it if things didn't go as expected.  What do you think?

Oh, oh.  I see there is another possibility for the file that stores the desktop state between sessions ---

PMAIL > MAIL > (name of mailbox with problem) > IMC-3E8 > DESKTOP.PDC

 I really need help from someone who understands how this program works better than I do.

According the Hans' Guide to Filenames and Extensions, FOLSTATE.PM is rebuilt on startup from HIERARCH.PM.  There is also a DESKTOP.PM0 that may play a role in this.

If you don't use trays in the folder list then I would be inclined to backup and then delete all three of these files.  Pegasus Mail will rebuild them.  DON'T do this if you use trays in the folder list because that structure will be lost by the rebuild of HIERARCH.PM.

IMC* indicates IMAP.  I believe there is an IMC* directory for each IMAP connection.  I don't believe anything in these directories should affect the Pegasus Mail startup desktop but I don't know much about IMAP, especially if an instance of Pegasus Mail only uses IMAP.

[quote user="jficklin"]Euler, thanks for your reply.  As I mentioned in my post as was able to change the "folders" window fonts to an appropriate size and the change persisted between sessions.  It is the closing of the folders window that should, but does not persist between sessions.
[/quote]

Well, this sounds strange as every window state is saved as soon as Pegasus Mail closes. BTW, the file that holds these windows' states is STATE.PMJ and I highly recommend you do not try to edit or delete it as it also carries lots of other information related to folders and windows. If you think this is your only option, then shut down Pegasus Mail and rename it to say STATE.PMJ.OLD and re-start Pegasus Mail with fingers crossed. Chances are that it may restore it to a default state but folder related rules (to name one) will be lost.

Maybe you could check if this file — or the Windows folder were it resides — attributes may be interfering with Pegasus Mail normal procedures. Another thing to consider is antivirus action. If your antivirus scans \PMAIL directory and sub-directories it may be interfering with this file update, maybe even others as well. Try turn your AV solution off for some time, make the change you need and close Pegasus Mail to see if it sticks.

Below a screenshot of my Pegasus Mail main window. Note that the Pmail open folder "Pegasus Mail Community" was used to test the Open/Closed Pmail folder state. If Pmail is closed with that particular folder opened (as in the screenshot) it will appear open on Pmail's next start, and NOT otherwise. Sorry if the image is badly shaped. I did it in a hurry.[;)]

I'm also pointing the Folders pane because I often get confused with Folders (the Pmail pane), Pegasus Mail folders and Windows folders. I try to refer to the prior with a capital ef always, and to qualify the others to avoid misunderstandings.