All Questions
98 questions
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Trace type convergence of the Laplacian on the box to the Laplacian on $\mathbb R^d$
Let $-\Delta \colon H^2(\mathbb R^d) \to \mathbb R^d$ be the (negative) Laplacian on the full space and $-\Delta_L$ the Laplacian acting on $L^2([-L,L]^d)$ with some boundary conditions making it self-...
- 61
1
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0
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98
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$(\lambda I-A)^{-1}-(\lambda I-B)^{-1}$ compact implies $\sigma_\text{ess}(A)=\sigma_\text{ess}(B)$
Suppose $H$ is a Hilbert space and $A$, $B$ are two adjoint operators on it (not necessarily bounded), satisfying $D(A)=D(B)$.
Question: If $\exists \lambda\in \rho(A)\cap\rho(B)$ such that $(\lambda ...
- 775
1
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0
answers
101
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If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?
Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...
- 63
2
votes
1
answer
237
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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 ...
- 1,171
3
votes
2
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294
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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}...
- 1,171
5
votes
2
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625
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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 ...
- 364
2
votes
1
answer
137
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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
\...
- 1,219
0
votes
1
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184
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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
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0
answers
151
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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 $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 101
2
votes
1
answer
195
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Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$
I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
- 229
6
votes
1
answer
575
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Spectrum of the complex harmonic oscilllator
Let
$$
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\...
- 501
1
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0
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76
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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 $...
- 307
1
vote
1
answer
119
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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
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0
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210
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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 ...
- 91
3
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0
answers
214
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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?
$$ V(f) (x) =\...
7
votes
1
answer
413
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 536
4
votes
1
answer
201
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Spectrum Cauchy-Euler operator
A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations
We consider the operator
$$(Lf)(x) = \...
- 536
0
votes
0
answers
109
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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
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0
answers
53
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 ...
- 563
1
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158
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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
3
votes
1
answer
107
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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 ...
- 91
1
vote
1
answer
119
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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
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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 ...
- 617
3
votes
0
answers
322
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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. ...
- 536
4
votes
1
answer
213
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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 ...
- 536
2
votes
0
answers
101
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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^...
- 536
1
vote
0
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71
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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:...
- 167
1
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1
answer
111
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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
5
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1
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229
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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 ...
- 620
4
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1
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301
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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 ...
- 620
5
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0
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515
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=\...
- 903
0
votes
0
answers
122
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
286
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) \| ...
- 1,175
1
vote
1
answer
136
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 $\varepsilon$-...
- 798
-1
votes
1
answer
246
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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 $A$ ...
- 167
4
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1
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161
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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
11
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1
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486
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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 ...
2
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0
answers
92
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First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
- 173
1
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1
answer
153
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Spectral properties of operators mapped to zero by some polynomial
Let $T$ be a bounded operator on a Banach space $X$ and suppose that there is a non-constant polynomial $p$ such that $p(T) = 0$. It seems to be well known that the spectrum of such an operator ...
- 381
0
votes
2
answers
244
views
Spectrum of a Markov kernel acting on $L^2$
Let $P$ be a Markov kernel on a measurable space $(E,\mathcal E)$ admitting an invariant probability measure $\pi$. $P$ acts on $L^2(\pi)$ via $$Pf:=\int\kappa(\;\cdot\;{\rm d}y)f(y).$$ The invariance ...
- 167
4
votes
0
answers
2k
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Eigenvalues and spectrum of the adjoint
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...
- 3,873
5
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1
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151
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Existence of operator with certain properties
I am curious to know the answer to the following question:
Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated ...
user69109
1
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0
answers
99
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Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$
I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
- 167
0
votes
0
answers
32
views
Spectral measures of a family of parameter-dependent self-adjoint contractions on an $L^2$-space
I have a self-adjoint linear contraction $A_g$ on an $L^2$-space of the form $$A_gf=\int\gamma(f,g),$$ where $\gamma$ is Lipschitz continuous and $g$ is an a priori fixed function. Assuming $1-A_g$ is ...
- 167
2
votes
1
answer
568
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Understanding a proof about limit of a sequence of open sets
We are reading a proof about the following limit
\begin{equation}\tag{1}
\lim_{n \to \infty} \sigma_1(T_n)= \sigma_1(T),
\end{equation}
where $T:D(T) \subseteq H \to H$ and $T_n:D(T_n) \subseteq H \to ...
- 229
3
votes
1
answer
1k
views
Different definitions of a relatively compact operator
(Cross-post from Math Stackexchange, where some work has been done in the comments)
Let $T,K$ be unbounded operators on a Hilbert space $H$.
I've seen the following definition of a relatively compact ...
- 1,474
0
votes
0
answers
224
views
Show convergence of a sequence of resolvent operators
Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...
- 167
1
vote
1
answer
213
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Graph Laplacian Operator
Consider the linear operator $\mathbb{L} : L^2([0,1])\to L^2([0,1])$ defined by
$$
(\mathbb{L}f)(x) = \int_0^1 xy(f(x)-f(y)) \mathrm{d}y
$$
for all $f\in L^2([0,1])$ and $x \in [0,1]$. Is $\mathbb{L}$...
- 31
2
votes
0
answers
306
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Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum
I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
- 7,609
4
votes
1
answer
350
views
Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?
The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...
- 25.8k