Tagged Questions

Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

71 views

Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
160 views

Fact regarding Kirchhoff's Theorem

Question regarding Kirchhoff's Theorem: If $L(G)$ denotes the Laplacian of a graph $G$ then Kirchhoff's Theorem states that number of spanning trees in $G$ is equal to $(-1)^{i+j} \det L(i|j)$ ...
46 views

Matrix -tree theorem-How to understand the theorem

I am having trouble understanding Kirchhoff's Theorem. The statement I want to prove is that if $\lambda_1,\lambda_2,...,\lambda _{n-1}$ are non-zero eigen values of $L(G)$ then Number of ...
21 views

Inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$

I want to calculate the inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$ where, $w \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector and $\alpha$ is some ...
35 views

Compact operators form the only closed proper ideal of bounded linear operators

I am trying to understand the following proof in Trace Ideals and Their Applications by Barry Simon (Proposition 2.1): Let $\mathcal{J}$ be a two-sided ideal in $\mathcal{L}(\mathcal{H})$ containing ...
93 views

Can $e^{ax}$ be said to be the eigenfunction of the operator $\frac{d^{(n)}}{dx}$?

I'm gradually getting familiar with operators (as they are used in QM) and the terminology surrounding them, and I was wondering whether all the (to me) well-known operators have straight-forward, ...
21 views

26 views

Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
24 views

Spectral theorem for unitary operators T o F

Si $T$ es unitaria y $B$ es una base de $V$ formada por vectores propios de $T$ entonces $B$ es un conjunto ortogonal. If $T$ is unitary and $B$ is a basis for $V$ consisting of eigenvectors of $T$ ...
48 views

Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
35 views

29 views

Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator (...
42 views

39 views

Let $\mu$ be a finite, strictly positive measure on $\mathbb{R}$, and let $k$ be a measurable positive-definite kernel. Assume $k$ is bounded, and let $T:L^2(\mu)\rightarrow L^2(\mu)$ be defined by $$... 1answer 41 views Orthogonal projection onto the eigenspace of compact, self-adjoint operators. Let T be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that f\in H, ||f|| =1 and ||(T-3)f||\leq 1/2. Let P be the orthogonal projection onto the direct sum of all ... 0answers 5 views Classify sub C^*- algebras of \mathbb{C}^{2 \times 2} Apparently if A is a sub C^*- algebra of the complex n \times n matrices then we can characterize these subalgebras as block matrices.Now, for the case n=2 I was wondering if there is an easy ... 0answers 34 views Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum? Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it. 1answer 32 views The topology of the spectrum of a linear operator In general a spectrum of a linear operator has a decomposition into three parts: point spectrum, continuous spectrum and residual spectrum. What I'm interested in is the topology of these parts of ... 1answer 39 views On important functions relflecting spectral properties of Jacobi operators The spectral analysis of Jacobi (semi-infinite, tridiagonal) operators acting on \ell^{2}(\mathbb{N}) is deeply investigated. A crucial role is played by function m which is usually known as Weyl ... 2answers 38 views Clarification on point spectrum of an operator From my understanding, if \lambda is in the point spectrum, then \lambda is a complex number such that it satisfies the equation (T - \lambda I) x = 0. My confusion arises from problems like ... 0answers 52 views Isolated Eigenvalues on the Extensions. I asked this question on Mathoverflow http://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer ... 1answer 42 views \lambda is an eigenvalue iff spectral measure of \lambda is nonzero Let M be a normal operator on a Hilbert space and let E be the spectral measure of \sigma(M) (the spectrum of M). Show that \lambda is an eigenvalue to M \iff E(\{\lambda\})\not = 0. ... 1answer 78 views If 0\leq a \leq b and a is invertible, then b is invertible Let \mathscr A be a unital C*-algebra and let a,b\in \mathscr A such that 0\leq a \leq b and a is invertible. How to show that b is invertible? (0\leq a \leq b means that a,b is ... 1answer 46 views Finding the spectral decomposition of \Delta= \frac{d^2}{dx^2} [closed] What is the spectral decomposition of the operator \Delta= \frac{d^2}{dx^2} in (L^{2}(\mathbb R), dx)? Thanks you in advance 1answer 67 views Spectrum of right shift operator in weighted l2 sequence space Let l_2(a) be a hilbert space defined with following inner product: \langle x_n,y_n\rangle = \sum a^k x_k y_k. (It's a weighted sequence space with the weights \omega_i = a^i). It's elements ... 1answer 40 views The k-th derivative of the resolvent set I want to prove$$\frac{d^{k}}{dz^{k}}(zI-A)^{-1}=(-1)^{k}k!(zI-A)^{-k-1}$$I have the resolvent equation (zI-A)^{-1}-(\lambda I-A)^{1}=(\lambda-z)(zI-A)^{-1}(\lambda I-A)^{-1}, i.e.$$\begin{...
32 views

I am doing some research that relies on Hadamard product of two matrices bound, the most famous one that I encounter is : $\rho(A\circ B)=\rho(A)\rho(B)$ this seems to be trivial when I test it with ...
880 views

Spectrum of shift-operator

Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ...
Is the spectrum of the laplacian on $L^1(0,1)$ with Neumann boundary conditions known?