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I'm a bit confused regarding the definition of error detection. Let $H$ be a Hilbert space, $C$ a subspace, $P\colon H\to C$ the projection, and $E$ a linear operator on $H$. Consider these two statements:

(i) If $x,y\in C$ and $Ex=y$, then $y\in \mathbb{C} x$.

(ii) There exists $\lambda_E\in\mathbb{C}$ such that $PEP=\lambda_EP$.

Statement (i) is the naive meaning of error detection, while the second is the definition used. My issue is that these two statements are not equivalent(?)

E.g., let $A\colon C\to C$, $A\notin\mathbb{C}I$, and $B\colon C\to C^\perp$ both be linear and injective, and set $E=\pmatrix{A & * \\ B & *}$. Then $E$ does not satisfy (ii), but it does satisfy (i) trivially, since $Ex$ is never in $C$ unless $x=0$. To me it seems like (ii) is instead equivalent to the statement:

(iii) If $x,y\in C$ and $PEx=y$, then $y\in \mathbb{C} x$.

Can someone help me clear this up?

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