I left (by mistake) an expensive library book on the plane. I keep checking with the library to see if the airline returns the book. There is a slip on the book telling people that it is a library book and to mail it back to a specific address.

Am I naive thinking that the airline cleaning staff and lost and found will mail the book back to the library?

Sorry to be negative, but, yes I believe you are.
The cleaning staff will most likely either keep the book if it is interesting (to them) or bin it if it is not.

Did you try calling the airline? I lost a pair of sunglasses once on Skywest and they found it and held it for me at the airport. Of course, if another passenger snagged it or if the cleaning staff tossed it, you won't have much recourse.

I just found a library book in the boarding area recently.
I returned it to my local library and they said they would get it back to the library it came from...hundreds of miles away.

I got Lucky twice. The first time the plane left the gate but was taken to a hanger for a total cleaning, so they knew about it and relaid it to the people cleaning it and I was able to pick it up at the Baggage Office later that night.

The other time the plane was already in the air but they contacted the city to where it was going and I did get it back.

Of cause I realized the loss within a very short time span. I guess if its hours or days the worse the chance of seeing the item again. Of cause if its left in a pocket of the seat and someone finds it before the Staff has a chance to see if its there, Good-by Charlie most likely.

Then there were a couple of times that I thought I left something on the plane since I didnt have it with me, only to find out I either forgot in my car or somewhere else.

I left one on a flight a few months ago ... call the airline that day with all the info - it never showed up ... guess whoever found it must have liked it.

Bruce

A couple of weeks ago, I left my passport and other travel documents on the CX 340 arriving from HKG in AMS... the only problem was that the plane was already towed away from the gate to a remote spot when I found out standing in line at passport control.

Luckily they were able to retrieve my documents within two hours, only for me to find out that they stored my bags in a place far far away because I wasn't going to make my connection to LIS.

On the plus I was escorted by a lovely young woman who inevitably showed me a large part of the airport that one usually does not dare to access

I have left more stuff on airplanes than I'd care to admit, and I've never gotten any of it back. Sometimes I've done everything that can be done (filling out forms; calling both arrivial and departure gates; calling lost and found at both airports). There's always a chance an honest soul will take some time our of their day to return the book, but ... tls

I would return someone's library book if I found it. I found a USB flash drive and found the owner and mailed it back to them. I would hope someone would return the favor if I ever was in that situation.

I also left a library book on a plane (DL, I think) once. They called me to ask if they should mail it somewhere and, if so, where. As it happens, I was connecting back through the same airport in another few days so I asked them to just hold it for me to pick up when I went through.

However, you're depending on the kindness of strangers. Nice when it happens, but not necessarily the way to bet.

Left my driving license on UA that was returned (mailed to my home address)
Left glasses on BA they said they found it but never managed to get it back
Left few books on different airlines those never returned