Captain Ahab (a character from Melville's Moby Dick), according to the definition of the Tragic Hero as provided by Aristotle, weakly fits into the definition of a tragic hero.
According to Aristotle, the tragic hero is defined by five main characteristics.
1. A tragic hero is born of nobility.
2. A tragic hero possesses hamartia (a tragic flaw) which leads to his downfall.
3. A tragic hero experiences peripeteia (reversal of fortune forced because of the tragic flaw).
4. The actions of the tragic hero result in an increased self-awareness and knowledge.
5. The audience must feel pity, sympathy, or fear for the character.
According to this, Captain Ahab is not the typical tragic hero. He was not born of nobility. Instead, readers simply know he was orphaned.
Captain Ahab does possess a tragic flaw--the inability to give up on the hunt of the whale.
Captain Ahab never possesses increased self-awareness. Instead, his hamartia leads to his death.
The last characteristic of the tragic hero is for the reader, or audience, to feel pity, sympathy, or fear for the hero. This is the tricky aspect of naming Captain Ahab a tragic hero. Some readers may feel these feelings for him, and other readers will not.
In the end, Captain Ahab only qualifies as a tragic hero under one, maybe two, of Aristotle's characteristics. Therefore, one could easily state that Captain Ahab is not a tragic hero.