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### Why is every selfadjoint operator closed?

I've read this theorem multiple times, but never seen a proof: Every selfadjoint operator is closed. But it's always been stated without a proof. Is it somehow obvious? I can't see it immediately ...
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### Why is $T$ normal and idempotent? [duplicate]

Let $V$ be an inner-product space, finite-dimensional over $\mathbb{C}$. Let the operator $T:V\to V$ satisfy $$T^2 = \frac{1}{2}(T+T^*)\,.$$ I'd like to prove that $T$ is normal, i.e., $T^*T = TT^*$, ...
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### Adjoint of a matrix and inverse of a matrix

As everyone know that we can use a matrix $A$ to represent an operator $T$. The adjoint of a matrix $A$ is denoted as $A^*$, which takes complex conjugate of $A$ and then transpose. My problem ...
• 7,991
1 vote
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### if range $A^* \cap$ range $B^* = 0$ and $B^*A = 0$ then $\text{rank }(A + B) = \text{rank }A + \text{rank }B$

Problem. Let $A, B \in \mathcal{M}_{m,n}(\mathbb{C})$ such that $\text{range }(A^*) \cap \text{range }B^* = \{0\}$ and $B^*A = 0$. Prove that $\text{rank }(A + B) = \text{rank }A + \text{rank }B$. My ...
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