Identifying Projection Operators: Is Idempotence Enough?

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To determine if an operator is a projection operator, idempotence (P = P^2) is necessary but not sufficient alone. A projection operator can be expressed as a sum of outer products, and applying the spectral theorem reveals that the eigenvalues must be either 0 or 1. This leads to the conclusion that for a positive operator to be a projection, all eigenvalues must equal 1 or 0, confirming the idempotence condition. Therefore, while idempotence is essential, additional conditions regarding eigenvalues are also required to fully identify a projection operator. Understanding these criteria is crucial for correctly identifying projection operators in mathematical contexts.
kini.Amith
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If we are given an operator, say in matrix or outer product form, then how can we check if it is a projection operator? Is idempotence a sufficient condition for an operator to be a projection operator or are there any other conditions?
 
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A positive operator P is a projection operator iff P=P^2.

To see it note a projection operator has the form sum |bi><bi|. Square it and you get the same thing. Apply the spectral theorem to an operator P such that P=P^2 and we have sum pi |bi><bi| = sum pi^2 |bi><bi| which implies sum pi (1-pi) |bi><bi| = 0. Hence pi (1-pi) = 0 ie 1-pi = 0, pi =1.

Thanks
Bill
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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