All Questions
Tagged with hilbert-spaces sp.spectral-theory
55 questions
0
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0
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55
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reference request: conditions for pointwise and operator-norm convergence of kernel projections
At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
- 3,065
0
votes
0
answers
45
views
Discrete and continuous representation in Hilbert space
I’m interested in using laplacian (−Δ)
eigenfunction as a basis for H1(Rn)
. I know that in H1(Ω)
, Ω
bounded this can be done so I was wandering about H1(Rn)
.
Now let eλ
be an eigenfunction ...
- 63.9k
3
votes
1
answer
335
views
Book on Hilbert spaces, including non-separable
I am looking for a book that develops the theory of Hilbert spaces, including the spectral theorems and unitary representations, but includes non-separable Hilbert spaces in the main exposition. Any ...
- 869
0
votes
0
answers
220
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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 $A$ ...
- 31
1
vote
1
answer
771
views
A bounded operator $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$ [closed]
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 ...
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0
votes
0
answers
252
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 ...
- 3,065
2
votes
1
answer
237
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 ...
- 4,351
2
votes
2
answers
1k
views
Lebesgue integral with respect to vector measures?
Good evening,
I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense ...
- 1
20
votes
8
answers
12k
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Can a self-adjoint operator have a continuous set of eigenvalues?
This should be a trivial question for mathematicians but not for typical physicists.
I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...
- 41.1k
0
votes
1
answer
184
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 ...
- 12.6k
3
votes
1
answer
209
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 ...
- 1,311
0
votes
0
answers
41
views
Uniqueness of solution to abstract wave equation with unsigned energy
Let $H$ be a self-adjoint operator on a Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle$). Suppose the spectrum of $H$ in $(-\infty, 0)$ consists of only finitely many eigenvalues $\mu^2_k &...
- 481
1
vote
0
answers
62
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 $\psi_h(x)$...
1
vote
1
answer
89
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,
$$S = -\frac{1}{2\pi i} ...
- 6,035
5
votes
1
answer
1k
views
Left and right eigenvectors are not orthogonal
Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
- 12.6k
4
votes
1
answer
249
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Is the Sobolev space $H^1(\mathbb{R})$ contained in the domain of $(-\partial_x \alpha(x) \partial_x)^{1/2}$?
Let $\alpha(x) : \mathbb{R} \to (0,\infty)$ have bounded variation (BV) and suppose $\inf_{\mathbb{R}} \alpha > 0$. Consider the second order differential operator
$$H : =-\partial_x (\alpha(x) \...
- 24.2k
0
votes
1
answer
268
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 ...
-1
votes
1
answer
164
views
Closure of the point spectrum of an unbounded diagonalizable operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 12.6k
2
votes
0
answers
69
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Mathematical reason for scatter states being special?
In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics.
My understanding is, if $O \in B(H)$ is a self-...
- 569
3
votes
0
answers
214
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) =\...
2
votes
0
answers
145
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How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?
I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...
- 1,848
0
votes
1
answer
102
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 ...
- 9,650
3
votes
1
answer
497
views
Hilbert-Schmidt integral operator with missing eigenfunctions
I'm having some issues with the spectral decomposition of the integral operator
\begin{equation}
(Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}.
\end{equation}
Since
\begin{equation}
...
- 56
1
vote
1
answer
119
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 ...
- 12.6k
7
votes
3
answers
2k
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 ...
- 1,869
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votes
0
answers
57
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Isolated eigenvalues of "bipartite" operators
Please note: This is a reformulation of a previous question of mine. The old question has been already answered to, so I prefer asking a new one. However, it looked like the old formulation did not ...
0
votes
1
answer
152
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Detecting isolated eigenvalues from local spectral measures
Please note: This question has been edited after it became clear from Christian Remling's answer that the original formulation was far from what I really meant to ask.
Let $T\ne 0$ be a self-adjoint ...
- 24.2k
5
votes
1
answer
229
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 ...
- 18.7k
4
votes
1
answer
301
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 ...
- 18.7k
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 ...
5
votes
1
answer
159
views
For self-adjoint $A$ and $B$, when is $(A+iB)^*$ the closure of $A-iB$?
Suppose that I have two self-adjoint operators $A$ and $B$ such that $\mathcal{D}(A)\cap\mathcal{D}(B)$ is dense and $B$ positive. Then $A\pm iB$ (with domains $\mathcal{D}(A)\cap\mathcal{D}(B)$) are ...
- 13k
2
votes
0
answers
131
views
Does a spectral theorem exist for linear operator pencils?
I was wondering if a version of the spectral theorem (the projection valued measure case) holds for linear pencils of the form
$$
A-\lambda B
$$
where $A,B$ are self-adjoint on some Hilbert space $\...
- 223
2
votes
1
answer
92
views
On the dimension of the range of the resolution of the identity
I want to prove the following: Let $A,B$ be bounded self-adjoint operators in a complex-Hilbert space and $E_A(\lambda)$, $E_B(\lambda)$ its corresponding spectral resolutions, i.e.,
$$A=\int_{[m_A,...
- 42.8k
-1
votes
1
answer
246
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 $A$ ...
CommunityBot
- 1
8
votes
1
answer
393
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A question about comparison of positive self-adjoint operators
I have the following question but have no idea on its proof (one direction is trivial):
Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
$$\...
- 42.8k
1
vote
0
answers
67
views
Spectral theorems for generalized Hermitian matrices
Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...
- 4,547
4
votes
0
answers
2k
views
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
6
votes
2
answers
539
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
- 42.8k
2
votes
2
answers
872
views
Spectral theory in non-separable Hilbert Spaces
I am wondering about what can be said about the spectral theorem for unbounded, self-adjoint operators in a non-separable Hilbert space. There is a comment in this sense to the question "Does spectral ...
- 42.8k
1
vote
0
answers
53
views
Spectrum of a $1$-parameter family of symmetric linear operators
I am working with certain submanifolds of symmetric spaces and, using a construction in Terng-Thorbergson, we ended up in the following Hilbert space problem:
Let $H$ be a (real) Hilbert Space and $...
- 163
1
vote
1
answer
977
views
Regarding essential spectrum of the unilateral shift operator
This is with context to Example 4.10 in Section 11 of the book : A course in functional Analysis by J.B Conway. Let $\sigma_{le}(S)$ and $\sigma_{re}(S)$ denote the left and right essential spectrum ...
- 3,486
0
votes
2
answers
465
views
Spectrum equals eigenvalues for unbounded operator
Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...
- 219
3
votes
1
answer
214
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Non-point spectrum for diagonalisable self-adjoint unbounded operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 23k
5
votes
1
answer
379
views
Hilbert representation of a bilinear form
Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...
- 7,893
3
votes
1
answer
232
views
what is about the corresponding power series?
According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes:
periodicity ~ potential well or lattice (order)
lack of absolutely continued spectrum ~ Anderson ...
- 188k
0
votes
0
answers
68
views
$H$ self-adjoint with mass gap, $P≥0,Ω∈D(P),H+λP$ self-adjoint $⟹$ for $λ$ small, $H+λP$ has gap?
Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
- 221
6
votes
1
answer
1k
views
Is the sum of spectral projections a projection?
Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
$$P_{\{\lambda_1,...\lambda_n\}}=\frac{...
- 3,984
2
votes
0
answers
238
views
Examples for Markov generators with pure point spectrum
I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...
- 235
1
vote
1
answer
127
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Integral representation of joint projection valued measures
Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
- 201
0
votes
2
answers
2k
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Spectral decomposition of compact operators
Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...
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