2
votes
1answer
14 views

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$? ...
8
votes
2answers
99 views

Ways to calculate the spectrum of an operator

Friends, I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. ...
2
votes
0answers
61 views

Reference for simplicity of the principal eigenvalue of the Laplacian

i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in ...
1
vote
1answer
82 views

Book: Functional Calculus

Is there a good book that investigates in detail the various kinds of functional calculus? I'm having now some knowledge about unbounded operators and integration but I would like to understand ...
1
vote
2answers
61 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
9
votes
3answers
179 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
3
votes
0answers
50 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
3
votes
0answers
53 views

Sources on Inverse Spectral Theory

I follow the book "Inverse Spectral Theory" by J. Pöschel and E. Trubowitz in which functional analysis is very much involved. I am curious about another approach using more real analysis and theory ...
2
votes
1answer
66 views

Spectral theorem for $n$-tuples of selfadjoint operators

I need a 'good' reference to the following version of the Spectral Theorem: Given $n$ commuting selfadjoint operators on an infinite-dimensional Hilbert space, there exist a Borel measure $\mu$ on ...
1
vote
0answers
96 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
2
votes
0answers
37 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
10
votes
1answer
718 views

Importance of Toeplitz operators?

I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I ...
1
vote
0answers
39 views

Normalized Cuts and Spectra

I'm looking for a fleshed out proof of the following theorem. Theorem: Let $G=(V,\mathbf{W})$ be an undirected, edge-weighted graph with normalized Laplacian $\mathbf{L}_N$. Furthermore, let ...
6
votes
1answer
141 views

Do elliptic operators on Riemannian manifolds have a regularizing effect?

I'm working on my master thesis and need to handle some spectral theory of the Laplace operator on compact Riemannian manifolds and especially on the sphere. While investigating essential ...
4
votes
0answers
322 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
1
vote
1answer
97 views

References for spectral measures

I am trying to learn a little bit about the spectral theory of unbounded operators but the textbook we are using (Birman and Solomyak: Spectral theory of Self-Adjoint Operators in a Hilbert Space) is ...
4
votes
1answer
512 views

Spectrum of sum of operators on Banach spaces

Let $A$ and $B$ be two operators on a Banach space $X$. I am interested in the relationship between the spectra of $A$, $B$ and $A+B$. In particular, are there any set theoretic inclusions or ...
1
vote
3answers
297 views

references for the spectral theorem

Recently, I am thinking about the question in spectral theory. And I finally found that I need help with the properties of unitary operator. Its a consequence of the spectral theorem for the normal ...