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Spectral theorem for unbounded operators

Part of the Spectral theorem for unbounded operators states that if $A$ is a self adjoint unbounded operator and $B$ is a bounded operator such that $BA$ is contained in $AB$, then $B$ commutes with ...
Isaac's user avatar
  • 771
3 votes
0 answers
95 views

Commutator of $A\otimes I$ and $I \otimes B$ vanishes?

Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
Hugo's user avatar
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0 answers
69 views

Perturbation of one-parameter groups of unitary operators

Let $H$ be a Hilbert space and let $h$ be a fixed, densely defined, possibly unbounded, self-adjoint operator on $H$. Letting $B(H)$ denote the space of all bounded operators on $H$, it is well ...
Ruy's user avatar
  • 2,263
3 votes
0 answers
98 views

Algebra core for generator of Dirichlet form

This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
Curious's user avatar
  • 143
3 votes
0 answers
390 views

How to prove the polar decomposition of unbounded operators?

Let $ T $ be a closed, densely defined operator on a Hilbert space $ H $. Then there exists a positive self-adjoint operator $ A $, $ D(A)=D(T) $ and a isometric operator $ V:R(A)\to \overline{R(T)} $ ...
Luis Yanka Annalisc's user avatar
3 votes
0 answers
201 views

'Local' commutativity of self-adjoint operators

Preamble Two (unbounded) self-adjoint operators $A, B$ on a Hilbert space $\mathcal{H}$ are said to (strongly) commute if the unitary groups they generate commute or equivilantely if all the ...
Photographer's user avatar
3 votes
0 answers
74 views

A question about a theorem in 'Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators'

I have asked this question on MathSE and someone advised me to ask it here. The link is . I'm studying the paper Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators ...
aeei.w.1995's user avatar
2 votes
0 answers
111 views

Everywhere-defined unbounded operators between Banach spaces

In this post, it is said that there are no constructive examples of everywhere-defined unbounded operators between Banach spaces; every example furnished must use the axiom of choice. This seems like ...
Damalone's user avatar
  • 151
2 votes
0 answers
261 views

When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?

Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$: $$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
H A Helfgott's user avatar
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2 votes
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91 views

How to show $ |(Bx,x)|\leq (Ax,x) $ for any $ x\in D(A) $ here?

On the Hilbert space $ H $, $ A $ is a non-negative self-adjoint operator and $ B $ is a symmetric operator. Let $ D(B)\supset D(A) $, where $ D(A) $ and $ D(B) $ are definite domain for $ A $ and $ B ...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
654 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
user270619's user avatar
2 votes
0 answers
144 views

Extensions of symmetric unbounded operators

I saw it claimed that every symmetric operator on a Hilbert space $H$ can be extended to a self-adjoint operator on some potentially larger space K. But I seem to be able to prove from this that every ...
Isaac's user avatar
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2 votes
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395 views

On strong resolvent convergence of unbounded operators

Given a sequence of unbounded self-adjoint operators $\Delta_N$ defined on some fixed domain of the Hilbert space $L^2(\mathbb{R})$ converging to i*identity in the sense that: for all $\phi\in\...
Kris's user avatar
  • 63
2 votes
1 answer
486 views

Common eigenvector of commuting unbounded operators

Let $T$, $S$ be two self-adjoint linear operators on a Hilbert space $\mathcal{H}$ with pure point spectrum. Then $T$ and $S$ commute if and only if they have a complete set of common eigenvectors. ...
Tim S's user avatar
  • 21
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0 answers
40 views

Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e., \begin{align*} g=\sum_{i=1}^n{x_i\textbf{1}_{...
Travis Mccormick's user avatar
2 votes
0 answers
363 views

Inverse and implicit function theorems with domain

I have seen an author use the Implicit Function Theorem for a map whose second partial derivative has a bounded inverse, but is unbounded. The map itself is not defined on an open set, but only on a ...
Benoît Kloeckner's user avatar
2 votes
0 answers
171 views

Operator theory of initial-value ODE problems

The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces. In contrast to that, the theory of ...
shuhalo's user avatar
  • 5,327
2 votes
0 answers
114 views

non-closed weak graph limit of symmetric operators

Hi Everyone, I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
Tlas's user avatar
  • 21
2 votes
0 answers
242 views

Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of $$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$. I also assume that $a(x)...
RadonNikodym's user avatar
2 votes
0 answers
137 views

Invariant linear manifolds for multiplication by the independent variable in L^2 (R)

In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-...
rtwmartin's user avatar
1 vote
0 answers
385 views

Densely defined and closed operator

Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be ...
John's user avatar
  • 85
1 vote
0 answers
153 views

Maximal domain of an unbounded linear operator in a weighted Hilbert-space

Let's consider the following (unbounded) linear operator. (So called transport operator in some context.) $$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
kumquat's user avatar
  • 185
1 vote
0 answers
205 views

Convergence in the resolvent sense and spectral properties

Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$. If $T_k\to T$ in the norm-resolvent sense, then for any $(a,b)\subset \mathbb R$ with $\{a,b\}\cap \sigma(T)=\emptyset$, ...
Watanabe's user avatar
  • 111
1 vote
0 answers
167 views

Common core for unbounded operators

Suppose that $\mathcal H$ is a Hilbert space representing some physical system, $H$ is the Hamiltonian for the system, and $A$ is some observable for the system, that is, some unbounded self-adjoint ...
Isaac's user avatar
  • 771
1 vote
0 answers
277 views

Adjoint for a non-densely defined unbounded operator on a Hilbert space

Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...
Max Schattman's user avatar
0 votes
0 answers
62 views

"Trade-off" between bound on the function and on the spectrum for functional calculus in spectral theory

Let $A$ be a self-adjoint (unbounded) operator on a separable Hilbert space $H$. From the following form of spectral theorem, we may define a functional calculus by $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
Ma Joad's user avatar
  • 1,755
0 votes
0 answers
377 views

Densely-defined operator with closed range: conditions for operator closed

Suppose we have Banach spaces (or Hilbert spaces) $X$ and $Y$, and a densely-defined linear operator $A : \operatorname{dom}(A) \subset X \rightarrow Y$ that is densely-defined and with closed range. ...
shuhalo's user avatar
  • 5,327