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13
votes
2answers
268 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 ...
11
votes
1answer
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 ...
9
votes
3answers
185 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
1answer
334 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 ...
8
votes
2answers
865 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 ...
7
votes
2answers
141 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 ...
7
votes
1answer
189 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 ...
5
votes
2answers
291 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 ...
4
votes
5answers
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 ...
4
votes
2answers
318 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' : ...
4
votes
3answers
284 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 ...
4
votes
3answers
275 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 $$ ...
4
votes
2answers
618 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 ...
3
votes
1answer
169 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 ...
3
votes
0answers
541 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 ...
2
votes
3answers
1k 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 ...
2
votes
2answers
223 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
1answer
114 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 ...
2
votes
0answers
88 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 ...
2
votes
0answers
178 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
0answers
96 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
3answers
518 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 ...
1
vote
3answers
473 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 ...
1
vote
1answer
94 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 ...
1
vote
1answer
225 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 ...
-2
votes
2answers
588 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 ...