'has a well-established reputation' is correct and usable in written English. You can use it when referring to something (e.g. a person, a company, a business, etc.) that has a long-held reputation. For example: The company has a well-established reputation for providing excellent customer service.
Florida has a well-established reputation as a swing state.
Diving: Dive Dominica, (888) 414-7626 and www.divedominica.com, has a well-established reputation and is convenient.
All of that said, Carmelo Anthony has a well-established reputation for being a shoot-first star, one who dominates the offense and who has never placed much emphasis on passing.
Among them, Rob Portman, who spent 12 years in the House before serving in the Bush administration and now seems likely to win an open Senate seat from Ohio, has a well-established reputation for working across the aisle.
This is particularly true for newer universities, who may not have a well-established reputation".
In response Mr Bashir described Palin as having a "well-established reputation as a world class idiot", before going on to cite the story of an incident where a slave was made to defecate in another's mouth.
Beckmann, one of the great German artists of the 20th century, had a well-established reputation for his harshly expressive canvases, characterized by rigidly defined blocks of color and a carved quality that evoked the Northern Gothic style.
Awesome tool! I started using it one year ago and I never had to look for another app
Ha Thuy Vy
MA of Applied Linguistic, Maquarie University, Australia