All Questions
Tagged with sp.spectral-theory operator-theory
145
questions
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
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
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
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
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
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 $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 151
3
votes
0
answers
160
views
On the spectrum of Fokker–Planck with linear drift
The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
0
votes
0
answers
100
views
Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?
Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...
- 195
3
votes
1
answer
153
views
On a compact operator in the plane
Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$
and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...
- 3,987
1
vote
0
answers
64
views
The spectral measure of the integral of an operator-valued function with respect to a projection-valued measure
Let $\mathcal{H}$ be a Hilbert space, $T$ a bounded self-adjoint operator, and $F:\left[a,b\right]\to\mathcal{B}\left(\mathcal{H}\right)$ such that for any $t\in\left[a,b\right]$, $F\left(t\right)$ is ...
- 11
2
votes
1
answer
189
views
Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$
I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
- 229
6
votes
1
answer
560
views
Spectrum of the complex harmonic oscilllator
Let
$$
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
1
vote
1
answer
112
views
Regarding variation of spectra
I have been reading the article The variation of spectra by J.D Newburgh. in this article and all related reference/ articles, the term 'variation of spectra' keeps coming in, but I nowhere find a ...
- 27
0
votes
0
answers
159
views
Reed-Simon Vol. IV: Question regarding convergence of eigenvalues
I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
- 71
3
votes
0
answers
174
views
Extended adjoint of Volterra operator
Let $V$ be a Volterra operator on $L^2 [0,1]$.
Does there exist a nonzero operator $X $ satisfying the following system
$VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator?
$$ V(f) (x) =\...
3
votes
1
answer
208
views
Reference request for spectral theory of elliptic operators [closed]
I want to learn the spectral theory of linear elliptic operators in bounded and unbounded domains in $R^n$, in particular for Laplacian and Schrodinger operators. Please suggest me some reference.
I ...
6
votes
1
answer
320
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 496
4
votes
1
answer
184
views
Spectrum Cauchy-Euler operator
A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations
We consider the operator
$$(Lf)(x) = \...
- 496
0
votes
0
answers
95
views
The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
2
votes
0
answers
52
views
A question about the choice of a special harmonc spinor
Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be ...
- 483
1
vote
0
answers
154
views
Relation between the spectrum of 𝐷𝐴 and 𝐴 where 𝐴 is a bounded linear self-adjoint positive operator and 𝐷 is a constant diagonal positive matrix
Let $D$ be a $3\times 3 $ constant real positive-definite diagonal matrix and $\Omega\subset\mathbb{R}^3$ be a bounded lipschitz domain.
Denote by $(L^2(\Omega))^3$ the set of square integrable ...
- 63
0
votes
1
answer
97
views
Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces [closed]
Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded ...
- 63
2
votes
1
answer
97
views
Finiteness of Schatten $p$-norm of truncated free resolvent
Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$.
For $p \geq 1$, let ...
- 71
2
votes
2
answers
379
views
Does this operator have a continuous, localized eigenfunction with negative eigenvalue?
I am looking at a class of operators
$$
L[f](x)=af_{xxxx}-bf_{xx}+\frac{d}{dx}(\delta(x)f_x)
$$ , a<0,b<0,
on the real line, where $\delta$ is Dirac-delta.
I am interested in ruling out the ...
1
vote
1
answer
99
views
Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$
In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested ...
0
votes
0
answers
100
views
Methods to find the spectrum of an operator
Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$?
Here is the setting I'm wondering about: consider ...
- 153
0
votes
0
answers
67
views
Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 525
3
votes
0
answers
313
views
Heat equation damps backward heat equation?
In a previous question on mathoverflow, I was wondering about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
- 496
4
votes
1
answer
184
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
- 496
2
votes
0
answers
101
views
Strongly continuous semigroups on weighted $\ell^1$ space
Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$
Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
- 496
1
vote
0
answers
69
views
Show that the Laplacian on these domains is isospectral
Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...
- 131
1
vote
1
answer
99
views
Sum of positive self-adjoint operator and an imaginary "potential": literature request
To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
- 13
7
votes
3
answers
1k
views
Essential spectrum of multiplication operator
Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its essential spectrum. Is there any way to explicitly compute its ...
- 119
5
votes
1
answer
178
views
Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research
In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete ...
- 580
4
votes
1
answer
248
views
Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus
In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim ...
- 580
4
votes
0
answers
418
views
Learning from eigenvalues of Hilbert-Schmidt integral operator
Do eigenvalues of the Hilbert-Schmidt integral operator determine the underlying measure up to translation, reflection and rotation?
Details: Suppose we have a measure $\mu$ on a Euclidean space $X=\...
- 649
0
votes
0
answers
101
views
Isolated points of the spectra of self-adjoint operators on Hilbert spaces
Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$.
I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
1
vote
1
answer
616
views
A bounded operator $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$
Theorem: Let $T$ be a bounded self-adjoint operator on a complex infinite-dimensional Hilbert space $H$. Then $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$.
Proof: If $T$ is compact ...
- 173
17
votes
1
answer
2k
views
Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?
Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
- 333
2
votes
1
answer
80
views
On the dimension of the range of the resolution of the identity
I want to prove the following: Let $A,B$ be bounded self-adjoint operators in a complex-Hilbert space and $E_A(\lambda)$, $E_B(\lambda)$ its corresponding spectral resolutions, i.e.,
$$A=\int_{[m_A,...
- 173
1
vote
1
answer
235
views
On a limit for the resolvent norm
Let $H$ be a complex, infinite dimensional, separable Hilbert space. Fix any two nonzero operators $A,B \in B(H)$ such that $B$ is not a scalar multiple of $A$. It is well known that:
$$ \| R_A (z) \| ...
- 1,155
3
votes
2
answers
207
views
A relation between norm and spectral radius for some matrix operators on Banach spaces $\ell^{p}$
Let $A=(a_{i,j})_{i,j=1}^{\infty}$ be a semi-infinite matrix with real entries. Suppose further that $A$ and $A^{T}$ (matrix transpose) represent bounded operators on $\ell^{p}$ for $p\geq1$. Denote ...
- 2,168
1
vote
1
answer
104
views
Uniform boundedness of resolvents on the imaginary axis
Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C_{0}$-semigroup. Suppose that in a $\varepsilon$-...
- 748
-1
votes
1
answer
220
views
Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators
Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
- 131
4
votes
1
answer
153
views
Elliptic estimates for self-adjoint operators
Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$
Let $T$ be a densely defined and closed operator ...
- 192
4
votes
1
answer
303
views
When is rank-1 perturbation to a positive operator still positive?
Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
- 610
11
votes
1
answer
417
views
Is the spectrum of a "self adjoint" operator real on $\ell^p$?
There might be an obvious answer to the question, but it doesn't come to mind.
Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
7
votes
1
answer
360
views
Spectrum of "classical" operators
Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
- 385