If A and B are positive transformations on a finite dimensional inner product space such that...

 

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If A and B are positive transformations on a finite dimensional inner product space such that A ≤ B, does it follow that A² < B²? How about √A≤ √B? If A and B are positive transformations on a finite dimensional inner product space such that A ≤ B, does it follow that A² < B²? How about √A≤ √B?