All Questions
Tagged with sp.spectral-theory fa.functional-analysis
367
questions
0
votes
0
answers
22
views
Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\mathrm{div}g(x)\nabla$
...
0
votes
0
answers
89
views
Self-adjoint operator with pure point spectrum
Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true:
A has pure point spectrum (i.e., the ...
3
votes
2
answers
158
views
How to diagonalize this tridiagonal difference operator with unbounded coefficients?
Problem: I have a self-adjoint operator in $\ell^2(\mathbb{Z})$ which acts as
$$T g(x)=q^{-2 x -3/2} g(x+1)+(1+q) q^{-2 x-1} g(x)+q^{-2 x +1/2} g(x-1),$$
and I am looking to diagonalize it. The ...
- 1,660
3
votes
1
answer
112
views
Spectra of products variously permutated
Let $x,y$ be elements of a Banach algebra $A$ and $\lambda\in\mathbb C\setminus\{0\}$. If $\lambda-xy $ is invertible, then $\frac1{\lambda}\big[1+y(\lambda-xy)^{-1}x \big]$ is clearly an inverse of $...
- 55.5k
3
votes
1
answer
126
views
Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard
In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
2
votes
1
answer
148
views
On spectral calculus and commutation of operators
Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
- 833
0
votes
0
answers
83
views
Operator identity
Let $T:\mathcal{D}(A)\to\mathcal{H}$ be a unbounded, self-adjoint, operator with positive spectrum $\sigma(T)\subset [\varepsilon,\infty)$ for $\varepsilon>0$. Hence $T$ is bijective with bounded ...
- 833
1
vote
0
answers
120
views
Generalization of Borel functional calculus
[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus]
Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
- 85
3
votes
2
answers
179
views
Domain of spectral fractional Laplacian
Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
- 833
6
votes
0
answers
143
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
0
votes
0
answers
79
views
Reducing subspaces of unitary operators
Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
- 1
1
vote
1
answer
111
views
Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
- 227
5
votes
2
answers
378
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
- 228
3
votes
1
answer
195
views
Convergence of spectrum
Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$.
Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
- 1,171
2
votes
1
answer
65
views
Spectral threshold effect: examples
I know that the effect of homogenization can be treated as a spectral threshold effect. I want to know more examples of spectral threshold effects in mathematical physics.
2
votes
1
answer
107
views
Disturbance of self-adjoint operator
Assume that $ A $ is self-adjoint operator and $ B $ is a bounded self-adjoint operator. The definite domain of $ A,B $, denoted by $ D(A) $ and $ D(B) $ satisfies $ D(A)\subset D(B) $. Show that
\...
- 1,091
0
votes
0
answers
87
views
Applications of finite speed of propagation property
Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and
$$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
- 265
0
votes
1
answer
126
views
Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
- 63
3
votes
0
answers
108
views
Leibniz rule bound for the inverse of the Laplacian?
Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\...
- 2,331
0
votes
0
answers
193
views
Spectral theorem for commuting operators
Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
- 3,141
1
vote
0
answers
131
views
Reference on spectral theory of self-adjoint operators
I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self ...
- 13
3
votes
1
answer
92
views
First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?
Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by
\begin{equation}
Hf(x)=x^2f(x)-f''(x)
\end{equation}
for any smooth function $f$ compactly ...
- 2,331
3
votes
0
answers
106
views
Reference request: trace norm estimate
In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$
then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 151
0
votes
0
answers
58
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 $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 3,898
3
votes
1
answer
416
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 ...
- 118
0
votes
0
answers
56
views
Commutativity of spectral projection with linear opeartor
For $A \in L(X)$, with A a being a closed operator and $X$ is a Banach space (with bounded $\sigma(A)$), define
$\mathbb{P} = \frac{1}{2 \pi i} \int_\gamma R(\lambda,A) \, d\lambda$ to be the spectral ...
3
votes
1
answer
144
views
Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?
This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
- 1,243
0
votes
0
answers
135
views
About the proof of Lebesgue decomposition theorem for Hilbert spaces
Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
- 1,145
1
vote
0
answers
104
views
Uniqueness of Borel functional calculus for unbounded self-adjoint operators
I was reading these short notes on the Borel functional calculus where the author discusses the uniqueness property of this calculus for both bounded and unbounded self-adjoint operators.
When it ...
- 1,145
3
votes
1
answer
288
views
Spectral Radius and Spectral Norm for Markov Operators
My question concerns differences between the spectral radius $\rho$ and norm $\| \cdot \|$ of Markov operators in infinite-dimensional Banach spaces. This is far from my area of expertise, that is ...
- 540
8
votes
1
answer
239
views
Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
- 81
3
votes
1
answer
163
views
$\tau$-measurable operator
Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
- 45
1
vote
0
answers
46
views
$L^2$ norm of a kernel with a variable width
Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
4
votes
1
answer
155
views
Characters of algebra of Schwartz functions
Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$.
Question: Does there exist some character (non-zero multiplicative ...
- 960
1
vote
1
answer
75
views
Show $A_0 + S$ is invertible, where $S$ is the Riesz projection of bounded $A_0$
Let $A_0$ be a bounded linear operator on a Hilbert space $H$. Suppose $0$ is an isolated point of the spectrum of $A_0$. Let $S$ be the corresponding Riesz projection, namely,
$$S = -\frac{1}{2\pi i} ...
- 459
1
vote
1
answer
245
views
Question on possibility of uniquely defining the FRFT via certain properties
I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
- 181
2
votes
1
answer
189
views
Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$
I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
- 229
2
votes
0
answers
111
views
On Fredholm alternative for Neumann conditions
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases}
-\Delta ...
- 1,330
4
votes
1
answer
661
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 ...
- 53
4
votes
1
answer
101
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 ...
- 419
0
votes
0
answers
142
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 $A$ ...
- 203
1
vote
1
answer
177
views
Spectral perturbation theory of discrete spectra in presence of continuous spectrum
This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...
- 2,176
6
votes
1
answer
560
views
Spectrum of the complex harmonic oscilllator
Let
$$
H_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0.
$$ It is known that the spectrum of $H_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^*\}$. Now put
$$
(U_\mu \phi)(x)= e^{\mu\...
- 469
1
vote
0
answers
67
views
Spectral measure for a finite set of mutually commuting normal operators
The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway:
Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
- 245
0
votes
0
answers
90
views
Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$
Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define
$$
u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}).
$$
Question. What are necessary and ...
- 6,586
2
votes
0
answers
45
views
Right spectral gap of vector of two independent Markov chains
Let $(X_i)$ be a stationary Markov chain on $S$ (a potentially uncountable space with a Borel sigma algebra) with stationary distribution $\pi$ and transition kernel $P$. Let $(Y_i)$ be a stationary ...
- 203
0
votes
1
answer
137
views
Determine if an integral expression is in $L^2(\mathbb{R})$
Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...
4
votes
0
answers
197
views
Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
- 3,044
9
votes
0
answers
207
views
Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,324
0
votes
0
answers
36
views
Regarding significance of spectral variation under algebraic operations
I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...
- 27