Spectral theory is a study of generalized notions of operators, eigenvalues and eigenvectors.

learn more… | top users | synonyms (1)

0
votes
0answers
64 views

Mysterious Commutation in an Unbounded Operators Argument (Compact Resolvent) — Is there a typo?

While reading A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Chapter 1, Section 4, I came across a passage that I really do not understand, and I am trying to see if ...
0
votes
1answer
14 views

Showing T intertwines $D_T$ and $D_{T^*}$ using Spectral Theorem

Suppose $T$ is a contraction on a Hilbert space $H$ (separable, if you wish). $D_T=(I-T^*T)^{1/2}$ and $D_{T^*}=(I-TT^*)^{1/2}$. I want to show that $TD_T=D_{T^*}T$. I had done this before using a ...
2
votes
1answer
88 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
1
vote
2answers
126 views

Spectral Measures: Spectral Spaces (I)

Problem Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its ...
2
votes
0answers
53 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
1
vote
1answer
53 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
1
vote
0answers
56 views

On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
2
votes
1answer
181 views

Eigenvalues of tridiagonal matrix

on page 13 of the paper here there is a proof in theorem 4 that all eigenvalues of this tridiagonal matrix, which has strictly positive entries down the subdiagonals, are simple. Unfortunately, I ...
0
votes
0answers
23 views

Numerical range of closure of operator

Let $B$ be an unbounded densely defined and closable operator. If $\mathcal{N}(B)$ is the numerical range, what can be said about the numerical range of its closure $\overline{B}$?
2
votes
1answer
76 views

Multiplication operator on $L^2$ and spectral theorem.

Let's consider the multiplication operator by the independent variable in $L^2(\mu)$, where $\mu$ is a borel regular measure on $\mathbb{C}$: $Mf(z)=zf(z)$. I want to show that if $\phi$ is a borel ...
0
votes
3answers
57 views

Spectral Measures: Support vs. Spectrum

Given a complex Hilbert space $\mathcal{H}$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: ...
0
votes
1answer
44 views

Spectrum of multiplication operator by the independent variable in $L^2$

If $\mu$ is a regular Borel measure on $\mathbb{C}$ with compact support $K$, define $N_\mu$ on $L^2(\mu)$ by $N_\mu f=zf$ (the multiplication by the indipendent variable). An exercise in "Conway" ...
5
votes
1answer
92 views

Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

Let $\mathcal{H}$ be a complex Hilbert space, and let $A$ be a bounded operator linear operator on $\mathcal{H}$ with $\|A\| \le 1$. It is known that $\|p(A)\|\le \sup_{|z|=1}|p(z)|$ for all complex ...
0
votes
1answer
38 views

Spectral measure and commutativity.

I want to prove that if $A\in B(H)$ and $N\in B(H)$ is a normal operator, and $AE(\Delta)=E(\Delta)A$, where $E$ is the spectral measure given by $N$ and $\Delta$ is a Borel subset of $\sigma(N)$, ...
1
vote
0answers
23 views

Eigenvalues of Hankel matrices

Let $\mathbf{A}$ be a $4-$ dimensional symmetric matrix with real entries, whose elements are given as \begin{equation} \mathbf{A} = \left( \begin{array}{cccc} a & b & c & d \\ b & c ...
0
votes
2answers
37 views

Some doubts concerning spectral theory.

Probably I'm saying something wrong (that's why the conclusions are strange) so please correct me! There is the continuous functional calculus for a normal element $N$ of a C*-Algebra. This means ...
9
votes
1answer
132 views

The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
6
votes
0answers
323 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
0
votes
0answers
41 views

Stability conditions for a non-negative AR(k) model

Given the sequence $\vec{x}_{n} = [A\vec{x}_{n-1}]_+$ where $[.]_+$ denotes the ramp function, i.e., $[.]_+ = x$ for $x>0$ and $0$ otherwise. $A$ is a $k$-by-$k$ matrix given by $$ A= ...
0
votes
0answers
62 views

Kac's question 'Can one hear the shape of a drum' and Sunada method, a clarification

I'm reading the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
0
votes
1answer
40 views

A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
1
vote
1answer
54 views

an exercise about the spectrum of an element in Banach algebra.

An exercise of Banach algebra, section of spectrum has wanted the proof of this statement: Let $A$ be a Banach algebra and $x\in A$. Show that for every open set $U$ in $\mathbb{C}$ that contains ...
1
vote
0answers
36 views

Representations and mutually singular measures

I'm finding some difficulties with an exercise from Conway and I ask for some help in understanding it: "Let X be a compact space and let $\{\mu_n\}$ be a sequence of measures in X. For each $n$ let ...
2
votes
2answers
81 views

Why is the spectrum usually defined for operators between Banach spaces?

The spectrum of a linear operator $L: \mathcal{D}(L) \rightarrow \mathcal{X} $ is generally defined for $\mathcal{X}$ a Banach space (for example wikipedia on link above, or spectral decomposition on ...
0
votes
0answers
8 views

Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...
0
votes
1answer
41 views

In Hilbert-space theory, is there a name for an operator “erasing” a projection?

Let $L$ be a self-adjoint operator with discrete spectrum $S=\{\lambda_1 < \lambda_2 < \dots \}$ on a Hilbert space $H$ such that the spectral theorem holds, i.e. for any $F \in H$ we have the ...
5
votes
2answers
76 views

Necessary and sufficient conditions for when spectral radius equals the largest singular value.

One well known fact about matrix norms is the following: If $\lambda_1\geq \dots\geq \lambda_n$ are eigenvalues of a square matrix $A$, then: $$\frac{1}{||A^{-1}||} \leq |\lambda|\leq ||A||$$ If we ...
0
votes
0answers
29 views

Dense invariant domain stable under resolvent?

I have thought about the following problem: Let $A_1\dots A_n$ a family of (unbounded) essentially selfadjoint operators on some Hilbert space $\mathcal{H}$ and $\Phi\subset\mathcal{H}$ the maximal ...
0
votes
0answers
34 views

How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?

What is the maximum number of eigenvalues $\lambda < 0$ for the trigonometric problems?: $$ \begin{array}{c} -\frac{d^{2}f}{dx^{2}}=\lambda f,\\ ...
2
votes
2answers
91 views

Definition of continuous spectrum of a bounded operator

Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ ...
0
votes
1answer
58 views

Compactness of $(x_1,x_2,…)\mapsto(0,x_1,x_2/2…)$

I read that the linear operator in the Hilbert space $\ell_2$ defined by $(x_1,x_2,...,x_n,...)\mapsto(0,x_1,x_2/2,...,x_n/n,...)$ is compact. I wanted to prove it by proving that the image of the ...
2
votes
2answers
41 views

Finite number of eigenvalues outside circle $\|\lambda\|>\delta>0$

I know that, for any $\delta>0$, a compact operator $A$ defined on a linear variety of a Banach space has only a finite number of linearly independent eigenvectors corresponding to the eigenvalues ...
2
votes
1answer
58 views

Spectrum of integral operator

Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T. My attempt: First I can show that $T$ is ...
0
votes
1answer
45 views

Spectral Decomposition of Function-of-Normal-Operator

In Arveson's book A Short Course on Spectral Theory, on page 64 (section on spectral measures) the author mentions the usual spectral decomposition of a normal operator $N$ as $$N=\sum_{\lambda \in ...
1
vote
1answer
85 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
0
votes
2answers
141 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
1
vote
1answer
58 views

Spectrum of Verschiebung

I read that the shift operator $A:\ell_2\to\ell_2$, $(x_1,x_2,x_3,...)\mapsto(0,x_1,x_2,...)$ contains $0$ in its spectrum, and that's clear to me. It is also clear to me that it has no eigenvalue. ...
0
votes
1answer
56 views

Banach Algebras: Continuity of Inversion?

Context: This question is related to this thread: Spaces of Functions Given a topological space $X$ and a Banach algebra with unit $B$. Consider a continuous map $F:X\to B$ that is invertible ...
2
votes
1answer
53 views

Eigenfunctions and spectrum of $T:H \to H^*$ where $H$ is a Hilbert space

Let $H$ be a Hilbert space with dual $H^*$. Suppose $T:H \to H^*$ is a linear bounded symmetric operator. (We probably don't want to identify $H$ with $H^*$). Can we talk about the ...
1
vote
1answer
75 views

spectral decomposition of a bivariate function

Now I have a function $f=f(x,y)$, smooth and symmetric(i.e. $f(x,y)=f(y,x)$ everywhere), with arguments defined on a compact set: $(x,y)\in[0,1]\times[0,1]$. I'd wish to know if $f$ can be expanded ...
2
votes
1answer
74 views

Is a bounded Borel function of a normal operator normal

I am playing around with Borel functional calculus to try to understand it, and made the following argument: Let $T\in B(H)$ (bounded operator on Hilbert space) be normal. Let $f\in C(\sigma(T)) $ ...
4
votes
1answer
143 views

Spectrum of an integral operator.

For any $f\in C([0,1],\mathbb{R})$ set $$ Tf(x) = \int_0^1 [\min\{x,y\}\cdot f(y)]dy. $$ I have just proved that $T$ is a compact operator from $C([0,1],\mathbb{R})$ into itself. I would like to know ...
0
votes
1answer
35 views

Invariant functions on product of ergodic systems is determined by eigenfunctions?

Given an ergodic measure-preserving system $(X,\mathcal{B},\mu,T)$, the product system $(X\times X, T \times T, \mu \times \mu )$ need not be ergodic, in other words: It may have non-trivial invariant ...
2
votes
1answer
52 views

Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
1
vote
1answer
195 views

Riesz Functional Calculus vs. Holomorphic Functional Calculus

"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...
0
votes
2answers
66 views

Function Spaces: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
0
votes
3answers
96 views

Selfadjoint Operators: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
0
votes
0answers
27 views

Define a positive dot product in $\mathbb{R^3}$

Consider the matrix $A= \begin{bmatrix} k & k-1 & 0 \\ 1-k & 2-k & 0 \\ 2k-3 & 2k-1 & 2 \end{bmatrix} $ with $k \in \mathbb{R}$ and let be $f_a: \mathbb{R^3} \rightarrow ...
1
vote
0answers
26 views

How to derive the spectral projection operator in finite element method?

For a compact self-adjoint operator $T$, the spectral operator can be defined as follows: $$E(\lambda) = \frac{1}{2\pi i}\int_{\Gamma}(z-T)^{-1}dz$$. For a finite dimensional approximate operator ...
0
votes
0answers
39 views

Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...