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Do the domains of the two square roots of a positive (unbounded) operator coincide? [closed]

Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
Zoltan Fleishman's user avatar
1 vote
0 answers
104 views

Kernel representation of a power of (pseudo-)differential operator

Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ...
Mirar's user avatar
  • 350
15 votes
1 answer
474 views

Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?

The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a ...
Carlo Beenakker's user avatar
2 votes
1 answer
256 views

A question about Dirac operators

Let $D$ be a Dirac operator on spinor bundle $S$ over even-dimensional non-compact spin manifold $X$, $$ \left<s_1,s_2\right>_{L_2} = \int_X \left<s_1,s_2\right> \quad \forall s_1,s_2\in\...
Radeha Longa's user avatar
9 votes
0 answers
364 views

Geometric motivation behind the Fredholm module definition

If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
Max Schattman's user avatar
0 votes
2 answers
465 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 ...
Bas Winkelman's user avatar
2 votes
1 answer
134 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 ...
Dale Wintermann's user avatar
2 votes
1 answer
238 views

Why is index unchanged after applying functional calculus?

Suppose $D$ is the Dirac operator on a closed spin manifold $M$, with spinors $S$. One can take the functional calculus of $D$ with respect to the continuous function $f:\mathbb{R}\rightarrow\mathbb{R}...
geometricK's user avatar
  • 1,903
4 votes
0 answers
118 views

Smoothness in von Neumann algebra of measurable functions

Let $A=L^{\infty}(M)$ be an algebra of essentially bounded measurable function on manifold $M$. Let $D$ be a first order elliptic differential operator acting on some hermitian bundle $S$ over $M$ (...
truebaran's user avatar
  • 9,330
2 votes
0 answers
160 views

Hopf algebra translations of relations in operational calculus

Three particularly important reps of the exponential formula (cf. MO-Q) are the refined Lah polynomials (OEIS A130561): Exp[o.g.f.] = Exp[formal power series]$\; =\exp[\frac{1}{(1-a.x)}]$, umbrally ...
Tom Copeland's user avatar
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9 votes
1 answer
2k views

Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator. Examples: $\exp(\varepsilon \...
user34129's user avatar
7 votes
2 answers
807 views

When is a Pseudo-differential operator trace class or in Dixmier ideal?

Let's denote the set of all Pseudo-differential operators with symbol of “order” $d$ by $\Psi_d(M)$ and Sobolev space on $M$ by $H_s(M)$. It is known that If $P\in\Psi_d(M)$ Then $P$ extends to a ...
Asghar Ghorbanpour's user avatar
11 votes
2 answers
2k views

How "generalized eigenvalues" combine into producing the spectral measure?

Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of ...
Reda's user avatar
  • 333
5 votes
1 answer
807 views

Self-adjoint extension of locally defined differential operators

The following is well known. Given a symmetric differential operator, like $\partial_x^2$, defined on smooth functions of compact support on $\mathbb{R}$, $C_0^\infty(\mathbb{R})$, one can count the ...
Igor Khavkine's user avatar