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Questions tagged [self-adjoint-operators]

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To find a symmetric operator that is not essentially self-adjoint under some conditions.

There is a theorem saying: If a symmetric operator $A$ (in the Hilbert space $h$) satisfies $Ran(A+z) = Ran(A+\bar z) = h,$ for some $z\in\mathbb{C}$, then $A$ is self-adjoint. However, this ...
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Conditions for Boundedness of Spectral Measures of Perturbations of Self-Adjoint Operators?

Suppose $A$ is an unbounded self-adjoint operator in a Hilbert space $H$ with discrete spectrum $$\lambda_0 < \lambda_1 < \cdots$$ bounded below with lowest eigenvalue $\lambda_0$, lowest ...
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Weyl sequence for oscillator harmonic

Consider the operator $T=-\Delta_{\Bbb{R}^2}-(x^2+y^2)$. How can I construct a sequence $ (u_n)\in D(T)=\{u\in L^2(R^2); Tu\in L^2(R^2)\}$ such that $\|u_n\|=\sqrt{\int_{R^2}|u(x,y)|^2}=1,$ $u_n\to ...
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Different conditions for self-adjoint property

Consider the following differential equation $$f(x)\partial_ty(x,t)+\hat{M}y(x,t)=0\,,$$ where $\hat{M}$ is a differential operator with respect to $x$ and assume $y(x,t)=T(t)v(x)$. Then, $$f(x)T'(t)v(...
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To prove an inequality regarding to the resolvent of a self-adjoint operator.

I've been studying functional analysis and currently solving problems in "Mathematcial Methods in Quantum Mechanics With Applications to Schrodinger Operators" written by G. Teschl, but I have a ...
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A multiplicative operator is self-adjoint

I am doing the following problem: Let $(M,\mathcal{A},\mu)$ be a general measure space, $f:M\to \mathbb{R}$ be a measurable function. Define the operator $A_f:\mathrm{dom}(A_f)\to L^2(\mu)$ by $$\...
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Is this statement still true with a weaker condition?

Let $H$ be a complex Hilbert space and let $A:\mathrm{dom}(A)\to H$ be an unbounded symmetric operator with dense domain. Prove that $A$ is self-adjoint if and only if there is a $\lambda\in\mathbb{C}$...
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Using Self-adjointness to make an integral vanish

The intricacies of differential forms are still somewhat lost on me. Given some compactly supported scalar function $f$ on a compact Riemannian manifold $(M,g)$, I'm looking to find: $$\intop_{M}\...
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Spectral representation of compact self-adjoint operators on a Hilbert space H

Theorem 9.9-1 of Kreyszig's "Introductory Functional Analysis with Applications" states that if $H$ is a complex Hilbert space and $T : H \longrightarrow H$ is a bounded self-adjoint operator, then $T$...
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Normal operator with real spectrum is hermitian.

Consider a normal operator on a complex Hilbert space with its spectrum contained in the real line. Show that the operator is hermitian without using the spectral theorem. What I have tried- Since ...
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Question about notation for rewriting integral of Laplace Beltrami operator

I was just reading about the Laplace Beltrami operator, which is a linear, second order, self-adjoint operator on a general Riemannian metric $g_\mu\nu$ space. $$\Delta A_{\mu}=\nabla^{\alpha}\nabla_{...
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Find out if a operator is self-adjoint or orthogonal

Let $V$ be a $\mathbb{R}$ inner product space, and $B=\left \{v_1, v_2, v_3 \right \}$ basis of $V$, with $||v_i||=1$ $\forall i=1,2,3$, and $<v_1, v_2>=<v_1, v_3>=0$ and $<v_2, v_3>=...
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If $A+B\ge C$, can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?

Let $A$ and $B$ be two positive operators on a Hilbert space. $C$ is a positive operator with $A+B\ge C$. Can we find positive operators $A_1\le A$ and $B_1\le B$ such that $A_1+B_1=C$?
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Eigendecomposition of Self-Adjoint Operator with Non-Positive Inner Product

The spectral theorem (well, some version thereof) says that if $A$ is a self-adjoint linear operator on a (finite-dimensional) Hilbert space $H$, there exists a basis of $H$ consisting of eigenvectors ...
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Convergence of (unbounded) self-adjoint operators

I'm learning about the dynamical convergence (i.e, convergence of the unitary group associated with each operator) and resolvent convergence of (unbounded) self-adjoint densely defined operators. I ...
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Let A be a self-adjoint, compact operator on a Hilbert space. Prove that there are positive operators P and N such that A = P − N and P N = 0.

I'm having trouble approaching this problem. I'm totally unsure how to approach this problem. Here's what I've tried so far: If A is self-adjoint, then $(A)^*=A^*$ and $(AB)^* = B^* A^*$.
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Fredholm Alternative for Singular ODE

Consider the following inhomogeneous boundary value problem, $$t^2 u'' + tpu' +qu = f(t), \ t \in [-1,1], \ \ u(1) = \alpha, \ u(-1) = \beta,$$ where $p$ and $q$ are constants. I would like to ...
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Proof of a linear algebra lemma for Cohn-Vossen's theorem

For the proof of Cohn-Vossen's rigidity theorem I need to prove the next lemma (can be found in Montiel-Ros's Curves and Surfaces page 218): If $\Phi$ and $\Psi$ are two definite self-adjoint ...
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Dimension of the null space of a compact perturbation of a self-adjoint operator

Let $L$ and $L_0$ be unbounded self-adjoint operators on $L^2(\mathbb{R})$ such as $L$ is a compact perturbation of $L_0$. I was wondering if it is possible to deduce an upper or lower bound on the ...
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Adjoint relation: transpose or conjugate transpose?

If an adjoint identity reads $$\boldsymbol{v} \cdot (\mathsf{C}\boldsymbol{x}) =(\mathsf{C}^+\boldsymbol{v})\cdot\boldsymbol{x},$$ where the adjoint vector $\boldsymbol{v}$ and the original vector ...
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Showing $(1-x^2)u''-xu'+9u=x^3$ is formally self-adjoint

I want to show $$(1-x^2)u''-xu'+9u=x^3 \ \ \ \ \ \ -1\leq x\leq 1$$ can be written in formally self-adjoint form. My attempt: Calculate the integrating factor, $$\exp\left(-\int \frac{x \ dx}{1-x^...
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Express in terms of $E$ a self-adjoint operator $T$ such that $T^2 = I+E$

I was trying the following problem: Let $V$ be a finite dimensional inner product space. Let $E: V \to V$ be an orthogonal projection onto some subspace of $V$. Express in terms of $E$ a self-adjoint ...