Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

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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 ...
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97 views

Doubt about integration over a curve

I'm reading a paper and I have a doubt: Suppose we have a compact Riemannian manifold and an eigenfunction of the Laplace operator $\Delta u=-\lambda u$. Suppose in addition that we have the following ...
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113 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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219 views

Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...
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341 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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100 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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91 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
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49 views

Is it possible for an operator to have only one eigenvalue in this case? - in need of a proof

First of all i have to state that i am a newcommer to spectral theory so please take it easy on me :). On lectures our professor derived this equation: \begin{align} \underbrace{\psi ...
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425 views

Recovering a Matrix knowing its eigenvectors and eigenvalues

Given the eigenvalues and eigenvectors of a matrix $R^{n\times n}$ is that possible to recover the same matrix from smaller matrices $R^{(n-1) \times (n-1)}$ where one of its eigenvalues and ...
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290 views

Continuous spectral value of right shift operator $\ell^2(\mathbb{N})$

Let $T:\ell^2(\mathbb{N} \to \ell^2(\mathbb{N})$ be the operator that sends$(x_1,x_2,x_3,...) \to (0,x_1,x_2,x_3,....)$. I want to show that $\lambda = 1$ is in the continuous spectrum. To approach ...
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317 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 ...
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123 views

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 ...
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227 views

A solution of $-y'' + q(x)y= \lambda y$

Could you help me with the following problem (from Poschel and Trubowitz)? I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with ...
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144 views

When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
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84 views

Time-independent magnetic Schrodinger equation

Let $B$ be a magnetic field, let $\mathbf{a}$ be a vector and let $\Psi$ be the wave function. If $\mathcal{H}(B) = (-i\nabla + B\mathbf{a})^2$, where $\nabla$ is the gradient, then the ...
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36 views

In which cases the spectrum of an operator contains only eigenvalues?

Let $X\neq \{0\}$ be a complex normed spaces (not necessarily finite-dimensional) and $T:D(T)\subset X\to X$ a linear operator (not necessarily bounded). I would like to know under what conditions can ...
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121 views

Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
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46 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 ...
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27 views

Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
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52 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 ...
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249 views

spectral integral

I am learning spectral integration for my summer. I am stuck at a point. Having got hold of a spectral measure, we define the spectral integral of a simple function as usual and then approximate any ...
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83 views

SVD, infinite matrices and normal operators from a function

I'm trying to understand the behavior the Singular Value Decomposition on a deeper level, and why it might give a particular result. Take the function $$ f(x,y) = \frac{1}{(1+2x+y)^2} $$ and ...
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In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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86 views

Compute the continuous spectrum of an unbounded operator in $L^2(\mathbb{R}^2)^2$.

In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily ...
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138 views

Spectra of a multiplication operator on $C^0(\overline\Omega)$

Let $\Omega \subset \mathbb{C}$ open and bounded, $X = C^0(\overline{\Omega})$ (vector space of continuous complex-valued functions). For an $y \in X$ with $y(\overline{\Omega}) = \{\lambda \in ...
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364 views

If $S$ and $T$ are commuting, normal operators, then $ST$ is normal

If $S$ and $T$ are commuting, normal operators, then $ST$ is normal That says it all, but let me be more specific. (By the way Wikipedia says this: "The product of normal operators that commute ...
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85 views

Asymptotics of Riemann-Lebesgue type integral

How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$, $$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$ Note that $\lambda$ ...
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146 views

Find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$

I need to find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$ in $L_2[0,\pi]$. I know that this operator is self-adjoint, so its residual spectrum ...
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187 views

Construct a multiplication operator which has dense point spectrum

By a multiplication operator here we mean an operator $$Af(t)=m(t)f(t), \qquad f \in D(A)=\{x \in L^2(\mathbb{R} \mid m(t)f(t) \in L^2(\mathbb{R})\}$$ where $m$ is a Borel measurable function on ...
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587 views

Continuous spectrum can shrink to an isolated point

Let $A$ be a bounded linear operator in a Hilbert space $H$. I had the misconception that the continuous spectrum of $A$ would necessarily have some "continuous" appearance: an interval, a union of ...
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34 views

Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
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83 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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58 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 ...
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interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
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Spectrum of the Orr Sommerfeld equation

The Orr Sommerfeld equation is as follows $$\psi''-k^2 \psi - \frac{U''}{U-c}\psi=0$$ where $\psi(y)$ is a complex valued function on $[0,2\pi]$ satisfying Dirichlet boundary conditions ...
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67 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
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65 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
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114 views

Application of the spectral decomposition theorem to PDE

In the compact version, there is many application of the spectral decomposition of a bounded self-adjoint operator: Sturm-Liouville, spectre of the laplacian,... But for the general version which ...
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209 views

Proof of the spectral theorem for normal operators from two lemmas

I have the following lemmas that I can prove: Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant Let $T$ be a normal ...
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36 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 ...
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Calculus of Variations statement of a Singular Value Decomposition?

My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
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Dirichlet eigenvalue problem on the Hilbert cube

I'm trying to solve the Dirichlet problem for the Helmholtz equation \begin{aligned}-\triangle u & = & \lambda u, & x\in\Omega,\\ u & = & 0, & x\in\partial\Omega, ...
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What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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241 views

Determining the spectral representation of a operator

The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as: $$ T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda}, $$ where $\sigma(T)$ is the ...
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127 views

Diagonalization discrete laplace operator

How do i diagonalize the discrete laplace-operator $\triangle$ on $l^2(\mathbb{Z})$ (defined by e.g. $\triangle e_k = e_{k-1} + e_{k+1}$, with $e_{k}$ being the canoical basisvectors of ...
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115 views

$L^p$ norm of diagonal is $\leq$ Schatten $L^p$ norm

Let $A = (a_{ij})$ be an $n\times n$ symmetric matrix. Its Schatten norm is defined by $\|A\|_{S^p}^p = \sum_{j=1}^n |\lambda_j|^p$, where $\lambda_j$ are the eigenvalutes of $A$. I am trying to prove ...
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174 views

Borel - Caratheodory Inequality

If $f$ is a complex-valued function analytic on $\{z:\vert z \vert \leqq r \}$, then for $\vert z \vert <r $, $$ \vert f(z) \vert \leqq \frac{2\vert z \vert}{2-\vert z \vert} \sup\{\Re f(w): \vert ...
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107 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
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181 views

Examples for spectrum of an operator

Looking for easy-to-understand examples for the spectrum of an operator, preferably so that they exposed some special properties. The right shift is a nice example of an operator which does not have a ...
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187 views

Using Rayleigh Quotient to approximate the first eigenvalue of the Laplace operator on the unit disk

Let $D\subset\mathbb{R}^{2}$ unit disk, the first eigenvalue of the Laplace operator holds: $\lambda_{1}=\inf\left\{ \frac{\int_{D}\left|\triangledown ...