The unbounded-operators tag has no usage guidance.

**0**

votes

**1**answer

102 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 ...

**6**

votes

**1**answer

238 views

### Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation.
Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...

**0**

votes

**1**answer

93 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...

**-3**

votes

**1**answer

184 views

### Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ?
Let $H$=$L^2(\mathbb ...

**-1**

votes

**1**answer

70 views

### Infinitesimal generator is bounded [closed]

Consider a strongly continuous semigroup of bounded linear operators $S(t):X\to X$. The infinitesimal generator of $S(t)$ is the linear operator $A:D(A)\subseteq X \to X$ defined by
...

**0**

votes

**0**answers

65 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. ...

**1**

vote

**2**answers

186 views

### Sum of two unbounded self-adjoint operators

Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are ...

**0**

votes

**1**answer

342 views

### Operator theory of the Hessian

How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...

**4**

votes

**1**answer

163 views

### Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...

**2**

votes

**1**answer

129 views

### Eigenvalues and Compact Resolvent

For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...

**0**

votes

**1**answer

60 views

### Pullback via flow as operator group

Let $X$ be a vector field on a manifold $M$ that induces a complete flow $\Theta_t$. Then the operator family $\Theta_t^*$,
$$\Theta_t^*u(x) = u(\Theta_t(x))$$
is a strongly continuous semigroup of ...

**3**

votes

**0**answers

98 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 ...

**2**

votes

**2**answers

276 views

### A question on unbounded operators

Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:
Every densely defined operator $A:D(A)\to ...

**0**

votes

**1**answer

160 views

### Densely-defined unbounded operators with large support

Most densely-defined unbounded linear operators on Hilbert spaces have a very large domain. In fact, for a lot of natural operators the intersection of their domains are still dense.
Let us consider ...

**1**

vote

**1**answer

124 views

### Laplacian on space of measures

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with ...

**1**

vote

**1**answer

117 views

### Special form of unbounded operators on $L_2(\mathbb{R}_+, \mathcal{H})$

I have the following problem;
Fix a Hilbert space $\mathcal{H}$. Let $S \colon \mathrm{Dom}S \subset L_2(\mathbb{R}_+, \mathcal{H}) \rightarrow L_2(\mathbb{R}_+, \mathcal{H}) $ be a closed densely ...

**10**

votes

**3**answers

268 views

### Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?

Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...

**8**

votes

**2**answers

173 views

### why is this a sufficient condition for a domain to be a core of an unbounded operator?

Let $\alpha:\mathbb R\to U(H)$ be a strongly continuous action of the reals on some Hilbert space, and let $A=-i\frac d{dt}\alpha(t)|_{t=0}$ be its infinitesimal generator, so that ...

**2**

votes

**1**answer

130 views

### Self-adjointness of a perturbed quantum mechanical Hamiltonian specified in an infinite matrix form

Consider an operator $H$ on the Hilbert space $\ell_2$ given as an infinite matrix with two pieces, one diagonal and one arbitrary:
$H_{ij}=E_i\delta_{ij}+V_{ij}$. This has a physical meaning in ...

**7**

votes

**2**answers

532 views

### C*-algebraic representation of observables vs self-adjoint operators one

I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...

**13**

votes

**2**answers

399 views

### Perturbation of unbounded self-adjoint operators

In the paper "A CRITERION FOR THE NORMALITY OF UNBOUNDED OPERATORS AND APPLICATIONS TO SELF-ADJOINTNESS" by M. H MORTAD (http://arxiv.org/pdf/1301.0241.pdf), the author states the following theorem
...

**7**

votes

**1**answer

251 views

### Product of commuting nonnegative operators

Let $V$ be a real vector space with an inner product and $A,B : V \to V$ linear maps which are self-adjoint nonnegative-definite, i.e. $\langle Ax,y \rangle = \langle x,Ay \rangle$ and $\langle Ax,x ...

**2**

votes

**3**answers

929 views

### How to define Laplacian on $L_2$

This might be a dumb question, but I thought the Laplacian (classical) is defined for $C^2$ functions. How do we extend that to be a self-adjoint operator on all of $L_2$?
Is it the so called ...

**3**

votes

**1**answer

300 views

### Hormander's bracket condition for the adjoint of an operator

Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator
$$
L = \sum_{i=1}^k X_i^2 + X_0~.
$$
Here, I assume that Hörmander's bracket condition is ...

**6**

votes

**3**answers

473 views

### ordered exponential of unbounded operators

Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
...

**2**

votes

**0**answers

99 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 ...

**4**

votes

**3**answers

317 views

### Bounded operators and axiom of choice

In the article below, it is shown that the proposition "Every linear operator defined on a whole Hilbert space is bounded" is consistent with the axioms of ZF + a weakened version of the axiom of ...

**9**

votes

**2**answers

2k views

### What is a good reference that compact resolvent implies Fredholm operator?

Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i.e. $A$ can be ...

**4**

votes

**2**answers

391 views

### Transpose of unbounded operators between Banach spaces.

Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator
$L' : ...

**8**

votes

**1**answer

411 views

### When the adjoint of a hypoelliptic operator hypoelliptic

Assume, $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.
Recall that $L$ is a hypoelliptic differential operator, if for ...

**2**

votes

**0**answers

193 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 ...

**2**

votes

**0**answers

108 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 ...

**1**

vote

**3**answers

759 views

### Countability of eigenvalues of a linear operator

Is it true that every closed operator on a separable Hilbert H space only has countably many eigenvalues?
Or put the other way around, if I want to ensure that a (not necessarily bounded) linear ...

**2**

votes

**2**answers

242 views

### Smooth dependence of the spectrum on the operator

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter.
To clarify, suppose I have a 1-parameter family $T_h$ of ...

**-2**

votes

**2**answers

602 views

### Question on Linear Operators

Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad ...

**3**

votes

**0**answers

760 views

### Product of positive commuting operators

Let $A$ and $B$ be positive commuting bounded operators on a Hilbert space. It can be shown by functional calculus that $AB=A^{1/2}BA^{1/2},$ so that $AB$ is again positive. If $A$ and $B$ are not ...

**1**

vote

**1**answer

250 views

### Commutativity of pullbacks and the exterior derivative as an unbounded operator on $L^2$

Let $d_c, \delta_c$ be operators with domains $D(d_c) = D(\delta_c) = C_{c}^\infty(\wedge T^\ast M)$. We let $d_c$ be the usual exterior derivative on compactly supported smooth forms, ie., $d_c\omega ...

**5**

votes

**2**answers

799 views

### Nice Classes of Non-Closable Operators

The only thing I know about non-closable operators can be summarised as "they exist, but they're nasty, so let's not talk about them!" This seems to be the case with everyone else I've talked to. I'd ...

**14**

votes

**1**answer

1k views

### Essential self-adjointness of differential operators on compact manifolds

Let $L$ be a linear differential operator (with smooth coefficients) on a compact differentiable manifold $M$ (without boundary). Suppose $M$ is endowed with a smooth volume form (actually, a smooth ...

**2**

votes

**3**answers

2k views

### Infinite hermitian matrix

Suppose we have a finite square n x n matrix of complex numbers H that is Hermitian and skew-symmetric:
$H^\dagger = H$ and $H^T = -H$.
(T denotes transpose, $\dagger$ denote conjugate transpose. I ...

**5**

votes

**5**answers

1k views

### Measurable functions and unbounded operators in von Neumann algebras

How do you define unbounded measurable functions for a general von Neumann algebra?
For the commutative algebra $L^\infty(X,\mu)$, we can consider the space of all measurable functions that are ...