Articles of spectral theory

Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f”$ and $D(T)=H^2\cap H_0^1$. And $S$ on $L^2\left[0,\infty\right)$ as $Sf=-f”$ and $D(T)=H^2 \cap \left \{f\in H^2 | f'(0)=0 \right \}$. Find out the Spectrum of T and […]

Spectral Measures: Spectral Spaces (II)

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its probability measures by: $$\nu_\varphi(A):=\|E(A)\varphi\|^2$$ Introduce the pure-point space: $$\mathcal{H}_0(E):=\{\varphi:\exists\#\Lambda_0\leq\aleph_0:\nu_\varphi(\Lambda_0)=\nu_\varphi(\Omega)\}$$ Construct its normal operator: $$\varphi\in\mathcal{D}(N):\quad\langle N\varphi,\chi\rangle=\int_\mathbb{C}\lambda\mathrm{d}\langle E(\lambda)\varphi,\chi\rangle\quad(\chi\in\mathcal{H})$$ Regard its eigenspace: $$\mathcal{E}_\lambda=\{\varphi:N\varphi=\lambda\varphi\}:\quad\mathcal{E}(N):=\cup_{\lambda}\mathcal{E}_\lambda$$ Then one has: $$\mathcal{H}_0(E)=\overline{\langle\mathcal{E}(N)\rangle}$$ How to prove this? Reference This thread is related to: Spectral Spaces (I)

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two […]

How to find the Laplacian Eigen Values of the given graph

Find the Laplacian eigen-values of the of the graph on $N$ vertices whose edge set is given by $\{(i,i+1),1\le i<n\}$ and the edge $(1,n)$ . The answer is given to be $2-2\cos{\frac{2k\pi}{n}}$ . My try: I tried to proceed by induction. For the case $n=3$ ;I got the matrix as $L$= \begin{array}{cccc} 2& -1 &-1\\ […]

equality of two operators…

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two operators in $B(X)$ such that for every $T \in B(X)$ we have $\sigma(AT)=\sigma(BT)$. Show that $A=B$. Here $\sigma(A)$ denotes […]

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is an interval $[m, M]$ with $M>0$. Is the above statement correct? How can I prove it? Thank you!!

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the quadratic form $(B\psi, \psi)$: $$\min (\sigma(B))=\inf_{\psi \in H} \frac{(B\psi, \psi)}{\lVert \psi\rVert^2}, \quad \max(\sigma(B))=\sup_{\psi \in H} \frac{(B\psi, […]

Wave kernel for the circle $\mathbb{S}^1$ – Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^1)=\{n^2 : n\ \in \mathbb{N}^*\}$, the trace (It this relevant for the question?) as distribution is simply $$w(t)=\sum_{k \geq 1} […]

Eigenvalues of doubly stochastic matrices

There was a long standing conjecture stating that the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of regular $k$-gons anchored at $1$ in the unit disc for $2 \leq k \leq n$. Mashreghi and Rivard showed that this conjecture is wrong for $n = 5$, cf. Linear […]

Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1’\in\mathcal{A}’$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ Can it happen that its spectra really differ: $$\sigma(A)\cup\{0\}\neq\sigma(A’)\cup\{0\}$$ (The scenario is inspired by possible noncanonical unital extensions.) One might be tempted to conclude that they agree as: $$\langle\iota(\mathcal{A}_0)\cup\{1\}\rangle\cong\mathcal{A}_0\oplus\mathbb{C}\cong\langle\iota'(\mathcal{A}_0)\cup\{1’\}\rangle’$$ But there is a flaw as the example below […]