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# Questions tagged [sp.spectral-theory]

Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices

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73 views

### Multiplicity of Laplace eigenvalues and symmetry

Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence \begin{equation} 0=\lambda_0<\lambda_1\leq \...
121 views

### On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
42 views

### Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
186 views

### The definition of simple eigenvalue

This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer. I am ...
83 views

### Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?

Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...
63 views

### How do I show countable additivity of (proposed) spectral measure in the proof of the spectral theorem?

I'm currently writing a Bachelors thesis based on the following paper: Douglas, R., & Pearcy, C. (1970). On the spectral theorem for normal operators. Mathematical Proceedings of the Cambridge ...
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### Spectral universality of sample covariance from unit sphere

If $x_1,\dots,x_n\sim \mu$ are mean-zero iid samples in $R^d$ that are drawn from unit sphere, with covariance $E x x^\top = I_d,$ and $C_n:=\frac1n \sum_i^n x_i x_i^\top$ is the sample covariance, ...
79 views

### Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive ...
1 vote
106 views

### Quiver representations and the standard matrix decompositions

Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form: $$M = A D B$$ where $D$...
147 views

### On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$ and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...
1 vote
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### On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation \begin{cases} -\Delta ...
475 views

### Left and right eigenvectors are not orthogonal

Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
99 views

### Uniform decay of operator norm for smooth family of operators

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...
111 views

### Eigenvalue multiplicity of tensor product of positive operator with itself

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
Let $\gamma_d$ be the standard Gaussian measure in $\mathbb R^d$, and let $W^{1,2}(\mathbb R^d,\gamma_d)$ be the Gaussian Sobolev space on $\mathbb R^d$. Fix $f_0 \in W^{1,2}(\mathbb R^d,\gamma_d)$. ...