How is Death of a Salesman a tragedy according to the definition(s) of tragedy?
The elements in Death of a Salesman that mark it as a tragedy are the ever declining fortunes of the hero and the ultimate demise of the hero. Elements that are contradictory of an Aristotelian Greek tragedy are that Willy Loman is a common man, a man of the people, and has no superior qualities by which to intensify the degree to which the audience can identify with and pity him, nor is the subject matter of a high "serious" nature; it too is commonplace.
In Aristotelian Greek dramatic (play) tragedy, the hero must combine good and evil qualities and have a reputation of renown that incorporates superior attainments. Pity for the hero and fear for impending catastrophe must grip the audience on the hero's behalf. Willy Loman matches none of these requirements. Further, the hero need not die.
In Elizabethan drama (plays), changes were made to the nature of the tragic hero. First, he was required to die in the end and, second, he fell due to a tragic flaw in his character as opposed to the fall of Greek tragedy which could be from several causes, like situational events or a mistaken decision. In earlier Medieval tragedy, the fall could be a reversal of fortunes on the wheel of fortune controlled by Dame Fortune.
Willy Loman seems to incorporate all these eras in that he is just a random man on the wheel of fortune without being particularly sympathetic to a broad audience who feel like they could--or wanted to--identify with him. Willy's fortunes turn on him as a man on the wheel of fortune...and he dies. His death offers a perverse parallel to the Greek resolution of tragedy in that the fallen hero would often learn something or become a better man. In Death of a Salesman, Willy's death liberates and frees his family because his insurance money pays off the house.