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2 votes
1 answer
237 views

On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
3 votes
1 answer
226 views

$\tau$-measurable operator

Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
3 votes
1 answer
192 views

Characters of algebra of Schwartz functions

Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$. Question: Does there exist some character (non-zero multiplicative ...
1 vote
0 answers
76 views

Spectral measure for a finite set of mutually commuting normal operators

The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway: Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
-1 votes
1 answer
164 views

Closure of the point spectrum of an unbounded diagonalizable 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 ...
3 votes
0 answers
160 views

Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras

I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
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 ...
8 votes
1 answer
393 views

A question about comparison of positive self-adjoint operators

I have the following question but have no idea on its proof (one direction is trivial): Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that $$\...
27 votes
0 answers
1k views

Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. What is an example of a non commutative ...
22 votes
5 answers
1k views

Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$

Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation $$ f(x+1) + f(x) = g(x). $$ We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
-1 votes
2 answers
640 views

Invariance of spectrum under conjugation

Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
3 votes
1 answer
214 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 ...
9 votes
0 answers
230 views

Using Property (T) to approximate invertible matrices

In the wikipedia article for Kazhdan's Property (T), there's an intriguing application: Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
4 votes
1 answer
161 views

Commutator representation of certain smoothing operators

I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...
0 votes
1 answer
220 views

Spectral decomposition function [closed]

Once I met a notation of "spectral decomposition function" (for a self-adjoint operator). No definition was given. Could someone give me a clue what can that be, cause I can't find this exact phrase ...
6 votes
0 answers
369 views

Paving conjecture for Toeplitz matrices

Let me first recall what is the so-called paving conjecture: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
5 votes
2 answers
491 views

Is independence meaningful for commutative $C^*$-algebras?

I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate. Let's say I have two self-adjoint operators on a Hilbert space and ...