The Batman Hush Logo originated from a 2002-2003 DC Comic book 12 part story line, written by Jeph Loeb, and penciled by Jim Lee, inked by Scott Williams and colored by Alex Sinclair. This story line introduces a mysterious stalker villain Hush, who seems intent on sabotaging Batman. Hush can be identified by his trench coat and face wrapped in bandages. It is eventually revealed that Hush is actually Thomas Elliott, who was Bruce Wayne's boyhood friend.
The Batman Hush Logo, probably the simpliest interpretation of the batman logo, is available on t shirts, a long sleeve shirt, tank top, sweatshirts, and hoodies for adults and kids.
To read more about Hush, go to this Batman Fandom site's page titled: Batman Hush.