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11 votes
2 answers
2k views

Spectrum of $L^\infty(X,\mu)$

Suppose that $(X,\Sigma,\mu)$ is a measured set with respect to $\sigma$-algebra $\Sigma$. Suppose that $L^\infty(X,\mu)$ is the set of all $\mu$-equal bounded $\Sigma$-measurable functions on $X$. ...
unknown is my last name's user avatar
3 votes
1 answer
228 views

Spectrum of a self-adjoint operator and spectral measures

Let $T$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, with spectrum $\sigma(T)$. For any $x,y\in \mathcal{H}$, denote by $\mu_{xy}$ the spectral measure of $T$ with respect to $x$ and $...
Maurizio Moreschi's user avatar
3 votes
1 answer
133 views

Restrictions on spectral measure

Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$ Here $\...
Qijun Tan's user avatar
  • 587
3 votes
2 answers
968 views

Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$ Is there any information ...
Student's user avatar
  • 617
2 votes
0 answers
131 views

Does a spectral theorem exist for linear operator pencils?

I was wondering if a version of the spectral theorem (the projection valued measure case) holds for linear pencils of the form $$ A-\lambda B $$ where $A,B$ are self-adjoint on some Hilbert space $\...
Mathmo's user avatar
  • 223
2 votes
0 answers
58 views

Absolute continuity of DOS measure for Schrödinger operators

Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure. I would like to ...
DDriggs's user avatar
  • 21
1 vote
0 answers
101 views

If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions Representation Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$. We call $\...
S-F's user avatar
  • 63
1 vote
0 answers
169 views

Generalization of Borel functional calculus

[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus] Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
oggius's user avatar
  • 95
0 votes
1 answer
268 views

Spectral Theorem, $AB = BA \implies B\Phi(f) = \Phi(f)B$

I'm studying the spectral theorem as appears in Reed and Simon's Functional Analysis. Assume we have constructed the continuous functional calculus for a self adjoint bounded operator $A$ on a ...
Mariah's user avatar
  • 181