# Questions tagged [unbounded-operators]

The unbounded-operators tag has no usage guidance.

89
questions

**1**

vote

**1**answer

126 views

### A consequence of the Min-Max Principle for self-adjoint operators

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...

**6**

votes

**2**answers

259 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 ...

**2**

votes

**1**answer

172 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 ...

**10**

votes

**1**answer

342 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....

**2**

votes

**0**answers

70 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)^{-...

**1**

vote

**0**answers

69 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 ...

**4**

votes

**0**answers

139 views

### 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 ...

**2**

votes

**0**answers

75 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 ...

**3**

votes

**0**answers

102 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 ...

**2**

votes

**1**answer

151 views

### When the adjoint of an unbounded operator on a Hilbert space coincides with the formal adjoint on its natural domain?

This is almost a copy of https://math.stackexchange.com/questions/3931318/when-the-adjoint-of-an-unbounded-operator-on-a-hilbert-space-coincides-with-the
I am trying to work with infinite matrices in ...

**0**

votes

**0**answers

53 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 $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...

**2**

votes

**0**answers

130 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\...

**0**

votes

**0**answers

31 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 $V_{j}(t)=V_{j}(t,\xi_{j})$...

**2**

votes

**1**answer

87 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 ...

**5**

votes

**1**answer

181 views

### Domain of $(-\Delta)^\alpha$, $\alpha \in (0,1)$

Is there any simple way to characterize explicitly the domain of fractional powers for a given operator? For example, the domain of Dirichlet Laplacian on a bounded nice domain $\Omega \subset \mathbb{...

**1**

vote

**1**answer

298 views

### Characterising closed range self-adjoint operators

Let $T:\mathrm{dom}(T) \subseteq H \to H$ be a densely defined, self-adjoint operator on a Hilbert spaces $𝐻$. In general the range of $T$ is not guaranteed to be closed. What tools are available to ...

**0**

votes

**1**answer

55 views

### An adjoint characterization of (unbounded) Fredholm operators

Let $\mathcal{H}_i$, for $i=1,2$, be Hilbert spaces, and $T:{\frak Dom}(T) \subseteq \mathcal{H}_1 \to \mathcal{H}_2$ a densely-defined closed operator. If the kernel of $T$, and the kernel of its ...

**2**

votes

**1**answer

172 views

### Closable unbounded operators and Banach space adjoints

For an unbounded operator $T:\mathcal{H}_1 \to \mathcal{H}_2$, if its adjoint $T^*$ is densely defined, then we know that $T$ is closable. What happens if we replace $\mathcal{H}_1$ or $\mathcal{H}_2$ ...

**5**

votes

**2**answers

320 views

### Unbounded Fredholms operators

Motivated by the situation of bounded Fredholm operators, I have the following question about "unbounded Fredholm operators".
Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces, and
$$
D: ...

**1**

vote

**1**answer

178 views

### Power of an infinitesimal generator of a $C_0$-semigroup in a Banach space

Let $A$ be the infinitesimal generator of a $C_0$-semigroup of linear operators in a Banach space. Let $n$ be a positive integer, $n\geq2$. Is $A^n$ closed?
Here (setting $A^1$ $:=$ $A$, and ...

**1**

vote

**0**answers

67 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 ...

**3**

votes

**1**answer

118 views

### The imaginary exponential of a tangent field on a manifold

If $M$ is a compact Riemannian manifold and $X$ is a tangent field, I am seeking to define the object $\exp {\mathrm i t X}$ for $t \in \mathbb R$, and I do not know how to do it.
One option was to ...

**0**

votes

**2**answers

248 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 ...

**3**

votes

**1**answer

107 views

### 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 ...

**5**

votes

**1**answer

409 views

### Unbounded version of continuous functional calculus

For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...

**3**

votes

**0**answers

66 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 ...

**2**

votes

**0**answers

160 views

### Common eigenvector of commuting unbounded Operators

Let be T,S two self adjoint linear Operator on a Hilbert Space $\mathcal{H}$ with pure point spectrum.
Then T,S commute if and only if they have a complete set of common eigenvectors.
This is the ...

**2**

votes

**0**answers

36 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}_{...

**4**

votes

**1**answer

393 views

### Commuting with an unbounded operator

Let $H$ be a Hilbert space. Let $A$ be a closed unbounded operator, and let $B\in B(H)$ be a bounded operator.
Definition:
$A$ and $B$ strong-commute if the partial isometry in the polar ...

**2**

votes

**1**answer

109 views

### Decomposition of the spectrum of an unbounded opeator [closed]

The Wikipedia article on spectral decomposition, see here
https://en.wikipedia.org/wiki/Self-adjoint_operator
says the following:
A self-adjoint operator A on $H$ has pure point spectrum if and ...

**1**

vote

**1**answer

280 views

### Dirichlet fractional Laplacian and zero boundary conditions

Does there exists a non-zero function $$f\in C_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C_0([0,1]) $, where $(-\Delta)^{\...

**7**

votes

**2**answers

396 views

### The von Neumann algebra generated by a non-closable operator

Let $H$ be a separable Hilbert space and let $M$ be a densely defined operator $\mathcal{D}(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\...

**7**

votes

**1**answer

255 views

### Does every integer map generate a von Neumann algebra of type I?

Consider a map $m: \mathbb{N} \to \mathbb{N}$ (we call it an integer map). Let $E_r$ be the set $m^{-1}(\{r\})$.
Let $H$ be the Hilbert space $\ell^2(\mathbb{N})$ and consider the densely defined ...

**1**

vote

**1**answer

235 views

### Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms?

Is it true that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?
This results is pretty easy and straightforward for $...

**5**

votes

**1**answer

1k views

### Commuting with self-adjoint operator

Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...

**4**

votes

**0**answers

115 views

### Bounded self-adjoint perturbation of a p-summable spectral triple

I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book 'Elements of Noncommutative Geometry' by Gracia-Bondía,Várilly and Figueroa.Now,after ...

**3**

votes

**1**answer

169 views

### Symmetric diagonalizable operators and self-adjointness

Given a densely defined symmetric operator $L$ on a Hilbert space $H$, which is also assumed to be diagonalizable, will there always exist a unique extension of $L$ to a self-adjoint operator?

**3**

votes

**2**answers

290 views

### Minimum eigenvalue of One-dimensional Schrodinger Operator

Consider the One dimensional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + V(x)
$$
Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $.
Now of course,the ...

**4**

votes

**1**answer

231 views

### Spectral growth of One dimensional Schrodinger Operator

Conside the One dimentional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + ( V(x) + E )
$$
Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $.
What is known ...

**4**

votes

**2**answers

797 views

### On the domains and extensions of unbounded operators

I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some ...

**4**

votes

**1**answer

145 views

### Scattering of relativistic particle by long-range potential

Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of ...

**7**

votes

**3**answers

445 views

### Notations for dual spaces and dual operators

I'm asking for opinions about the 'best' notations for:
1. the algebraic dual of a vector space $X$;
2. the continuous dual of a TVS;
3. the algebraic dual (transpose) of an operator $T$ between ...

**0**

votes

**1**answer

260 views

### Is the range of a weak convergent sequence also a weak convergent sequence?

Under which condition the image of a weakly convergent sequence (xn) by an unbounded linear operator T is a weakly convergent sequence (Txn) ?

**2**

votes

**0**answers

272 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 ...

**1**

vote

**1**answer

176 views

### Stochastic integral is a continous or closed operator?

The Setup
Let $\xi_t$ be a process adapted to the filtration $\mathfrak{F_t}$ of the semi-martinagale $X_t$, such that both are square integrable. Then is the map
\begin{align}
F_T: L^2(\mathfrak{...

**2**

votes

**1**answer

214 views

### Selfadjointness of hamiltonian with 1/x potential

Let us consider the Hilbert space $L^2([0,\infty))$ and operator $H=-\frac{d^2}{dx^2} + \frac{1}{x}$ on the domain of $C^{\infty}_0((0,\infty))$ (smooth functions with compact support away from $0$).
...

**2**

votes

**1**answer

61 views

### On extending a semigroup property

Let $T(t)$ be a $C_0$-semigroup on a Hilbert space $H$ with a generator $A$.
It is well known that for all $x\in H,$ we have: $ \int_0^t T(s)x ds \in D(A) $ and $ A\int_0^t T(s)x ds = T(t)x-x$.
How ...

**0**

votes

**1**answer

52 views

### Simplify the expression of $ T^+$ for an unbounded operator $T$?

For a negative unbounded operator $T$, what equals the operator
$$ T^+ = \left[\frac{1}{2}(|T| + T) \right]^{**},$$
where $|T|= (T^2)^{1/2}$ and $A^{**} $ is the minimal closed extension of an ...

**4**

votes

**1**answer

118 views

### For self-adjoint $T$ on $L^2(\mathbb{R}^n)$, when does $(1 + |x|)^{-1} (T - i \varepsilon)^{-1}(1 + |x|)^{-1}$ have a limit as $\varepsilon \to 0$?

Let $T : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be a possibly unbounded self adjoint operator. Let $R_\varepsilon$ denote the resolvent $(T - i \varepsilon)^{-1}$, $\varepsilon > 0$. Suppose that,...

**0**

votes

**1**answer

235 views

### Unbounded operator [closed]

Is the Laplacian operator $\Delta: C^2([a,b],\mathbb{R}) \to L^2(a,b)$ unbounded? Here $a,b \in\mathbb{R}$, $C^2([a,b],\mathbb{R})$ associated with $L^2$ norm.
If yes, how can we "modify" these ...