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### Range of restriction of a Hilbert adjoint operator

Let $V, \hat V$ be Banach spaces with $H, \hat H$ Hilbert spaces such that $V \subset H$ and $\hat V \subset \hat H$, both continuous, compact and dense injections. They have different norms. Suppose ...
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### a linear map without an adjoint

Define $W = \{(a_1, a_2,\cdots) : a_i \in \mathbb{F}, \exists N\in\mathbb{N}, \forall n \geq N, a_n = 0\},$ where $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$ and $W$ has the standard inner product, ...
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### Positivity of an eigenvalue of a composition of a differential operator and its formal adjoint

For $q:\mathbb{R}\rightarrow\mathbb{R}$ let $\delta$ be a differential operator of the form $$\delta=\frac{d}{dx}+q(x),$$ defined on some dense subspace of $L^2((0,1),d\mu)$, where $\mu$ is a finite ...
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### show that Norm of v+iT(v) equals to the Norm of v-iT(v)

So I have $V$ an inner product space above $C$ and a linear operator $T$ such as $T=T^*$ on $V$ I need to prove that: $$||v+iT(v)||=||v−iT(v)||$$ I tried to write it by definition, but I didn't get ...
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### Does $P \circ P =P$ and $\langle Px, y \rangle = \langle x, Py \rangle$ imply $P$ is linear?

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space and $P:H \to H$. Suppose that $P \circ P =P$ $\langle Px, y \rangle = \langle x, Py \rangle$ for all $(x,y) \in H^2.$ I ...
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### Adjoint of right shift operator on orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\ell^{2}(\mathbb{N})$

I'm sorry if this question is a duplicate. Suppose $(e_{n})_{n\in\mathbb{N}}$ is the usual orthonormal basis of $\ell^{2}(\mathbb{N})$. We can define an operator $v\colon H\to H$ by $ve_{n}:=e_{n+1}$....
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### Question concerning proof of unique existence of adjoint operator

Here, a proof is given on p. 50. However, one crucial moment is when they define $$T^{\star}k := y.$$ Okay, the next step reads: \langle h, T^{\star}\left( \lambda_1k_1 + \lambda_2k_2 \right)\...
On the space of $L^2([0,1])$ consider the following operator; $\Lambda u(x)= \int_0^xu(s)ds$ , I want to find the adjoint of this operator. \begin{align} (\Lambda u , g) & = \int_0^1\Big(\int_0^...
Consider $X\in\mathfrak{sl}_2(\mathbb{R})$, and an irreducible smooth representation $(\pi,V)$ of $SL_2(\mathbb{R})$. Assume $u,v\in V$, and let $<\cdot,\cdot>_P$ be the Petersson inner product, ...