# Questions tagged [unbounded-operators]

The tag has no usage guidance.

111 questions
Filter by
Sorted by
Tagged with
114 views

### $(S\otimes T)^{it}= S^{it}\otimes T^{it}$ for unbounded operators

Let $S,T$ be unbounded, closed operators in Hilbert spaces $H,K$. In that case, we can form the tensor product operator $S\otimes T$ on the Hilbert space $H\otimes K$ which is the closure of the ...
80 views

Let $\rho, \delta, m$ be real parameters such that $0\le \delta\le \rho\le 1, \delta<1$. The set $S^m_{\rho, \delta}(\mathbb R^{2n})$ is defined as the set of smooth functions $a$ on $\mathbb R^n\... 2 votes 0 answers 84 views ### How to show$ |(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 ...
104 views

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 \...
44 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 ...
216 views

### Diagonalizing selfadjoint operator on core domain

Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$...
405 views

### Takesaki II "Connes cocycle derivative"

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108: Why are the second and third ...
198 views

### If $A$ is a closed operator, is $A^k$ closed?

Let $A$ be a closed (densely defined) operator on a Hilbert space $H$. We define for a natural number $k$, the operator $A^k$ with its natural domain. Is $A^k$ closed?
112 views

### On the Fractional Laplace-Beltrami operator

I would appreciate it if a reference could be given for the following claim. Let $g$ be a Riemannian metric on $\mathbb R^n$, $n\geq 2$ that is equal to the Euclidean metric outside some compact set. ...
222 views

### Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book. Suppose ...
319 views

### Takesaki II Lemma 1.13: stuck in proof

Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"): Here, we associate with an ...
220 views

1 vote
128 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 $\{a,b\}\cap \sigma(T)=\emptyset$, ...
745 views

432 views

### Convergence criterion in the domain of an unbounded operator

Cross-post from math.sx. My question is somewhat close to this one, but the counterexamples given there do not apply here. Setup. Given a Hilbert space $\mathcal H$, a closed operator $A$ and a ...
209 views

### Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$

Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the ...
469 views

### A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner

$\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange https://math.stackexchange....
395 views

34 views

### What is the analog of the symmetrized Jacobi matrix for delay equations?

For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...
113 views

### Spectral representation of closed operators with finite spectral bound

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the ...
224 views

70 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 ...
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. ...
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}_{...