The tag has no usage guidance.

375 questions
Filter by
Sorted by
Tagged with
72 views

### Proving that eigenvalues of real self adjoint endomorphism are real

If $\mathbf{A}\colon V \to V$ is selfadjoint for some finite dimensional vector space $V$ over $\mathbb{C}$ then this follows immediately, since for any nonzero eigenvalue $\lambda$ and a ...
1 vote
41 views

I relate to this one question about corollary 4.10.2 pag. 198 of "Introduction to Hilbert Spaces - Debnath, Mikusinki" third edition, that states Let $A$ be a compact self-adjoint operator ...
1 vote
31 views

### Question related to a bounded operator in a Hilbert space.

The statement is "Let $A_n$ be a sequence in $B(H)$ and $A\in B(H)$ such that $\|A_n -A\|$ $\rightarrow 0$ and $n \rightarrow 0$ if $A_n$ is self adjoint then $A$ is also self-adjoint." I ...
1 vote
51 views

### Proof if $T \in L \left( V,V \right)$ is self-adjoint and b,c $\in \mathbb{R}$ such that $b^2 < 4c$ , then $T^2 + bT + cI$ is a positive operator.

I also have a hint: Use Cauchy–Schwarz inequality. I'm really stuck with this problem. I think that $T^2 + bT + cI \geq 0$ because of $b^2 - 4c < 0$ may help but I don't know how to relate it with ...
87 views

### Is self-adjoint operator in a Hilbert space is always positive operator?

Let me first define the self-adjoint operator. Let $A$ be a bounded operator in a Hilbert space $H$, then $A$ is said to be a self-adjoint operator if $A^*=A$. And $A$ is known as a positive operator ...
40 views

### Relation between resolvent set, self adjoint and open neighborhood

My professor gave me a list of exercises to complete in preparation for the exam ... but i can't solve this question. I have the feel that the solution is simple but in this moment I can't figure it ...
162 views

### Laplace-Beltrami operator is essentially self-adjoint on a bounded domain

I came across this lecture note The Poincaré inequality on domains on a webpage. In the first section, it claims that $L$ is essentially self-adjoint on $\mathcal{D}^\infty$. It is known that the ...
13 views

### How to show that the operator $(-\Delta)^{-1} : C^\beta(\overline{\Omega}) \to C^{\beta}(\overline{\Omega})$ is self-adjoint?

Well, I think that this operator can be seen as a constraint of the operator $(-\Delta)^{-1}:L^2(\Omega) \to L^2(\Omega)$ because $C^\beta(\overline{\Omega}) \subset L^2(\Omega)$ and by the regulary ...
24 views

### A basic question about the spectral theorem for self adjoint compact operator..

In the statement of this theorem which i state from the book of conway that if T is a compact self adjoint operator on a hilbert space then T have countable no of non zero eigen value ln and there ...
22 views

36 views

### A Geometrical Derivation of Quantum Mechanics Spin Operators

I'm trying to see if there is a way to geometrically derive a general form for the quantum mechanics spin operators. I'm trying to deduce their commutation relations without using any knowledge of ...
37 views

### Spectrum of general projection and orthogonal projection

I am trying to think about this, but I seem to be stuck. Suppose $P$ is a projection on a Hilbert space $\mathcal{H}$. If I am just talking about a general projection, where I only know that $P^2=P$, ...
37 views

### If $T^2$ is positive(that is $T^2$ is self-adjoint and $\langle T^2v,v\rangle\geq 0$) , when $T$ can be self-adjoint?

Suppose $V$ is a vector space with dimension $n$, I'm doubting that if operator $T^2$ is positive, then $T$ under some situations can be self-adjoint. However, I'm not certain about my reasoning. My ...
1 vote
63 views

### If the operator $P$ is positive and $T$ is self-adjoint, then is there exists a positive number $n$ such that $nP+T$ is positive?

$V$ is a finite-dimensional complex inner product space and $P, T\in L(V)$. If the operator $P$ is positive and $T$ is self-adjoint, then is there exists a positive number $n$ such that $nP+T$ is ...