In Woolf's "A Room of One's Own," what belief is implied in the "old gentleman's" statement that it is "impossible for any woman . . . to have the...

Simply put, the belief implied in the statement you reference is that women are inferior to men in all aspects of life, including the creation of literature. Since women are inferior to men (the old gentleman claims), they cannot produce anything as masterful as Shakespeare's plays. 

It is this belief that Virginia Woolf is primarily critiquing in her essay "A Room of One's Own." By employing impeccable logical reasoning, historical analysis, and literary examination, Woolf explores the rampant misogyny that has oppressed women through the ages and prevented them from pursuing any kind of independent activity, including the production of art and literature. Woolf ultimately agrees with the old gentleman, asserting that, back in Shakespeare's time, it would have been impossible for a female artist to have done what Shakespeare did. However, Woolf does not make this claim because she believes that women are inferior to men; rather, she says that a woman in Shakespeare's time could not have been a prolific artist because she would have been prevented from gaining employment, living independently, and using her time to create art. Instead, the patriarchy would have subjected her to cruel toil in the home and stripped her of the privilege of working and owning property. As such, by partially agreeing with the old gentleman's statement, Woolf brilliantly dismantles the implied misogyny behind it.