Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [self-adjoint-operators]

The tag has no usage guidance.

0
votes
1answer
39 views

Proof of strong limit $\displaystyle s-\lim_{\varepsilon \to 0} (I+i\varepsilon A)^{-1}=I$

Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$, I was wondering if there was an 'elementary proof', i.e. that doesn't use the full functional calculus, of the strong limit : $$\...
0
votes
1answer
16 views

Are Hermitian operators positive?

I can only find the proof for the reverse statement (i.e. here). However Nielsen Chuang Quantum Computation and Quantum Information p. 90 states the following: Suppose we define $$E_m = M^\...
2
votes
1answer
121 views

Lagrange multiplier term in Hamiltonian

My question is about a step in this paper: PhysRevB.65.165113 (X.G. Wen) or arxiv page 6. Or alternatively: PhysRevB.90.174417 or arxiv page 3. All papers on spin liquids and the projective ...
3
votes
0answers
63 views

What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
0
votes
0answers
40 views

Show that the operator associated to a spectral decomposition on a Hilbert space is self-adjoint

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
0
votes
0answers
14 views

Polar decomposition for closed operators

State and prove the polar decomposition for closed operators. My attempt: Let $A$ be a closed operator, we know $A^*A$ is self-adjoint. Let $W=(A^*A-1)(A^*A+1)^{-1}$. Then the polar decomposition for ...
1
vote
0answers
47 views

Some questions about the spectral composition of a nonnegative self-adjoint operator

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
0
votes
1answer
57 views

Is this Sturm-Liouville problem self-adjoint?

We are interested in determining whether the problem $\begin{cases}xu''-u'+u = \cos(x)\\u(0) = 0 \\ u(1) = u'(1)\end{cases}$ is self-adjoint. This is not a Sturm-Liouville problem, the corresponding ...
0
votes
1answer
17 views

self-adjoint operators $T,P$ and their kernel

Consider $T,P \in \mathcal{B}(H)$ to be self-adjoint operators, where $H$ is a Hilbert space. Show that $\ker(T^*T + P^*P) = \ker T \cap \ker P$. I feel like this should be easy to prove. I can prove ...
1
vote
0answers
32 views

Source for a particular proof of the spectral theorem.

Consider the following ''spectral decomposition'' for self-adjoint compact operators: If $T\neq 0$ is a self-adjoint compact operator on a Hilbert space $H$, then there exists a sequence $\{\...
0
votes
1answer
36 views

Show that an operator is symmetric but not selfadjoint.

I am stuck with the following exercise: Show that the operator $A= -d^2x$ with $D(A) =\{f \in L_2[0,1]:f,f' \in C[0,1] \,with\, f'' \in L_2,\, f(0)=f(1)=0 , f'(0)=f'(1)=0 \}$ Is symmetric but not ...
2
votes
1answer
58 views

A self-adjoint operator with essential spectrum={0} is compact

Does every self adjoint operator (on a Hilbert space) with essential spectrum={0} is a compact operator ?
1
vote
1answer
34 views

proof about self-adjoint operators $T,P$

Prove that for self-adjoint operators $T,P \in B(H)$ where $H$ is a Hilbert space ($B(H)$ is the space of all bounded functions from $H \to H$) and $T \leq P$ then $\|T\|\leq \|P\|$ . So by ...
0
votes
1answer
36 views

A self-adjoint operator without eigenvalues and with spectrum equal to {0}

Let $A$ be a self-adjoint operator on a Hilbert space $H$. We know that the spectrum of $A$ ( $\sigma(A)$) can be decomposed into an essential spectrum ($\sigma_{ess}(A)$) and a set of eigenvalues ($\...
3
votes
2answers
54 views

Strictly positive inner product for a pair of non-zero, positive operators.

Let $ A,B $ be non-zero positive operators on a infinite-dimensional separable Hilbert space $(H , \langle \cdot, \cdot \rangle)$. I am required to prove that there exists $u' \in H$ such that \begin{...
0
votes
1answer
69 views

RAGE theorem for absolutely continuous spectrum?

Given a (possibly unbounded) self adjoint operator $A$ acting on a Hilbert space $\mathcal{H}$, the RAGE theorem gives a characterisation of $$ \|P_{c}\psi\|^2 $$ where $\psi\in\mathcal{H}$ and $P_c$ ...
0
votes
0answers
14 views

Self-adjoint implies symmetry

I have found little statement that I'd like to prove. It's about: self-adjoin operator implies that this operator is symmetric. Definitions: Operator T: H-> H (where H-Hilbert's) is symmetric, if = ...
0
votes
1answer
29 views

Connection between selfadjoint and normal matices

Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint, such that $[A,B] := AB − BA = 0$ . Show that $C := A + iB$ is normal matrix. Could someone give me a hint on this problem ? I think that as ...
4
votes
3answers
58 views

Bounded and Self-adjoint Linear Operator and Its Inverse

Let $H$ be a Hilbert space and suppose that $A:H \rightarrow H$ is a bounded, self-adjoint linear operator such that there is a constant $c>0$ with $c\|x\| \leq \|Ax\|$ for all $x\in H$. Prove that ...
1
vote
3answers
84 views

We have a linear operator T. Show $T^2=Id$ implies $T=T^*$

We have a linear operator $T:V\rightarrow V $. V is a finite-dimension inner product space over the field of the complex numbers. Show $T^2=Id$ implies $T=T^*$. I've tried working with the inner ...
1
vote
0answers
25 views

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 ...
1
vote
0answers
27 views

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 ...
0
votes
0answers
55 views

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 ...
1
vote
0answers
24 views

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(...
1
vote
1answer
24 views

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 ...
1
vote
1answer
29 views

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 $$\...
0
votes
0answers
17 views

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 $\lambda\in\mathbb{C}$...
0
votes
0answers
7 views

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}\...
0
votes
0answers
29 views

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 $T$...
1
vote
1answer
53 views

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 ...
0
votes
1answer
28 views

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_{...
0
votes
1answer
56 views

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>=...
2
votes
2answers
29 views

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 $A_1+B_1=C$?
0
votes
1answer
63 views

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 ...
4
votes
0answers
44 views

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 ...
0
votes
1answer
108 views

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 $(AB)^* = B^* A^*$.
3
votes
1answer
79 views

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 ...
3
votes
2answers
90 views

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 ...
0
votes
0answers
36 views

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 ...
1
vote
0answers
74 views

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 ...
2
votes
0answers
48 views

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^...
1
vote
2answers
63 views

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 ...
4
votes
1answer
131 views

Show that $x$ is an eigenvector of $T$ with eigenvalue $\|T\|$

Let $H$ be a Hilbert space and let $T:H\to H$ be a bounded self-adjoint linear operator. Assume there exists $x\in H$ with $\|x\|=1$ and $|\langle Tx,x\rangle|=\|T\|$. Show that $x$ is an ...
2
votes
2answers
95 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle \...