What Does It Mean for Two Operators to Be Unitarily Equivalent?

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Two operators are considered unitarily equivalent if there exists a unitary operator, P, such that P^{-1}AP = B, where A and B are the operators in question. This concept parallels the definition of unitary equivalence in matrices, where two matrices are unitarily equivalent if they share the same eigenvectors, indicating they represent the same linear transformation in different bases. The discussion emphasizes the importance of unitary operators in understanding the relationship between different operators in functional analysis.

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Hi!
Could anyone please tell me the meaning of
Tow operators are unitary equivalent.
I tried Wiki but I did not get my goal.
 
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On this page: http://en.wikipedia.org/wiki/Similar_matrix, about "similar matrices", Wikipedia says that two matrices, A and B, are "unitarily equivalent" if and only if there exist a unitary matrix, P, such that P^{-1}AP= B. Two matrices are "similar" if there exist any matrix, P, such that AP= PB, with P not necessarily unitary. Two matrices are unitarily equivalent if and only if they have the same eigenvectors. That is the same as saying they represent the same linear transformation in different bases.
 
Thank you, But I wonder if this can be generalized to the operator case instead of matrices
 

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