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### self-adjoint operator and symmetric operator

we recently learned about self-adjoined operator with the formal definition $⟨Tv, w⟩ = ⟨v, Tw⟩$ for every $v, w$ in $V.$ In the other side we talked that self-adjoined can be represented as a ...
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### Concept of 2-variable function for operators on an $n$-dimensional inner product space

I'm reading the book "Finite-Dimensional Vector Spaces (2nd Ed)" by PR Halmos. The concept of a 2-variable function (or polynomial) for operators is introduced in Theorem 1 of Section 84 on page 171 ...
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### functional analysis( self-adjoint operators on a separable Hilbert space )

let $A_n$ and $A$ be self-adjoint operators on a separable Hilbert space, and let $A_nx$ converge to $Ax$ for any $x\in X$. Prove that for any continuous bounded $f$,$f (A_n) x$ holds and converges ...
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### Boundary Value ODEs Reference Book

Can someone suggest me a book on Boundary Value Problems in ODEs, which start from the general theory, and then go on to specialize for self-adjoint values? All the books I have found discuss the self-...
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### Is self-adjoint operator necessarily linear?

Let $(H, \langle\cdot, \cdot\rangle)$ be a Hilbert space and $P: H \to H$. In this answer, @gerw said that if $$\forall (x,y) \in H^2:\langle Px, y \rangle = \langle x, Py \rangle,$$ then $P$ is ...
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My teacher gave me a problem: I have two self-adjoint operators $A$ and $B$. Both of them are greater or equal than $I$, i.e. $A \geq I$ and $B \geq I$. I need to to prove that there is a positive ...
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### Tail of increasing convergent net of self-adjoint operators is bounded

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (...
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### Proof that compact self-adjoint operators have at least one non-zero eigenvector (using something analogous to min-max theorem)

I checked several texts on the spectral theorem for compact self-adjoint operators (like this (PDF) and this (PDF)) on Hilbert spaces. They all mention that the cluster points of the real eigenvalues ...
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### Show $A = \{ u \in S^+(E) \textrm{ | } \forall x \in K, \langle x, u(x) \rangle \leq 1 \}$ is a compact set

The problem Let $\left( E, \langle \cdot, \cdot \rangle \right)$ be a euclidean space of dimension $n$. Let $K$ be a compact subset of $E$ containing a basis $e = (e_1, ..., e_n)$ of $E$. We denote ...
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### What will be the matrix of an anti-selfadjoint transformation in an orthonormal basis?

I am a bit confused by this question. The statement of the question tells me that $(x|Ty) = -(Tx|y)$ and that $B$ is an orthonormal basis. Also, that the field is $C$. On my own, I have found that ...
Let $(E,\mathcal E,\mu)$ be a measure space and $A$ be a self-adjoint bounded linear operator on $L^2(\mu)$. Assume $Af\ge0$ for all $f\in\mathcal L^2(\mu)$ with $f\ge0$. Let $$c_1:=\sup_{\substack{f\... 0answers 32 views ### checking if an endomorphism is self-adjoint. Let V be an inner product space and let \alpha be an endomorphism of V. Is the endomorphism \alpha^*\alpha - \sigma_1 of V self-adjoint? Could anyone give me a hint on how to check this? 1answer 38 views ### Bounded self-adjoint linear operator is injective… Let T : H → H be a bounded self–adjoint linear operator on a Hilbert space H. Suppose the range R(T) is dense in H. Prove that T is injective. 1answer 63 views ### Sum of Shift Operators I have the operator T: l_2[0,\infty) \to l_2[0, \infty) defined by T=S_l+S_r. So T(x_0,x_1,x_2,x_3,...)=(x_1, x_2+x_0,x_3+x_1,x_4+x_2,...). I've shown that it's spectrum is a subset of [-2,2].... 1answer 45 views ### Does there exist a basis of \mathbb R^3 consisting of eigenvectors of T? True or false (and give a proof): There exists T ∈ L(\mathbb R^3) such that T is not self-adjoint (with respect to the usual inner product) and such that there exists a basis of \mathbb R^3 ... 0answers 28 views ### What is the self-adjoint operator of the sobolev semi-inner product \langle u', v' \rangle in the general case? I have a simple question that is bothering me. It is so common, that the answer should be easy to find, but I didn't. I am probably making it more difficult than it is. Consider a domain \Omega=(0,1)... 0answers 20 views ### Optimize the contractivity of a self-adjoint operator on a closed subspace Let H be a \mathbb R-Hilbert space and A\in\mathfrak L(H) be self-adjoint with$$\left\|Ax\right\|_H\le\left\|x\right\|_H\;\;\;\text{for all }x\in H.\tag1 Let $K\subseteq H$ be closed and ...
I need to prove that a self-adjoint operator $T \in \mathcal{B}(\mathcal{H})$ is an orthogonal projection if $\sigma(T) = \{0,1\}$. I know this means I have to prove $T$ is idempotent, meaning \$T = T^...