Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.
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2answers
138 views
Spectrum and point spectrum of this operator
Let $T\in \text{Aut}(\ell^2(\mathbb{C}))$ and $T(x)=(a_1 x_1, a_2 x_2,\ldots)$ where $a=(a_i)_i \in \ell^\infty(\mathbb{C})$. How can I easily see what is $\sigma(T)$ and $\sigma_p(T)$ (that are ...
2
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0answers
107 views
Examples for spectrum of an operator
Looking for easy-to-understand examples for the spectrum of an operator, preferably so that they exposed some special properties. The right shift is a nice example of an operator which does not have a ...
2
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1answer
194 views
Fourier transform physical meaning [closed]
What is the physical meaning of the Fourier transform expressed at the spectral density? Also, what is the relationship between the Fourier transform and the total energy of an oscillating system? ...
1
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1answer
74 views
Difference in sound between a string and a pipe
I am told that I can model the vibration of a guitar string of length $l >0$ by the following Sturm-Liouville equation
$$ -u'' = \lambda u \: \: \text{ on } [0,l],$$
with boundary conditions
...
1
vote
1answer
98 views
closed operator, projection
Let $A: D \subset X \to X$ be a closed linear operator. X is a Banach space. Furthermore we have $\gamma: [0,1] \to \mathbb{C}$, $\gamma$ is a $C^1$ curve and $\gamma \subset \rho(A)$, where $\rho(A)$ ...
2
votes
0answers
168 views
Spectral radius and positive definite of matrices
Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
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0answers
40 views
Compute Rayleigh quotient for ODE
I am trying to find Rayleigh quotient for this equation:
$u''(r) + [\frac{1-4n^2}{4r^2} + \lambda - 2n\beta -\beta^2r^2]u(r) = 0$, where $0 \le r \le 1$.
Is there any way to compute eigenvalue ...
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1answer
53 views
what is the largest eigenvalue of the average of non-negative matrices?
I have a set of square matrices $A_i \in \mathbb{R}^{n \times n}$ for $i=1,\ldots,N$, such that $[A_i]_{jk} \ge 0$ for all $i$ and coordinates $j,k$.
If the largest eigenvalue of each $A_i$ is ...
2
votes
1answer
32 views
Tensoring Spectral triples that are composed from Real algebras.
I have a misunderstanding that I am hoping is really quite trivial.
In connes standard Non-commutative geometry model of electroweak interactions he takes the algebra input in his finite spectral ...
3
votes
4answers
94 views
Eigenvalues of the “Laplacian” on [0,2$\pi]\subset\mathbb{R}$
for the dirichlet eigenvalue Problem on compact and connected Riemannian manifolds, the eigenvalues of the laplacian consists of a discrete sequence.
On the other hand, if we consider ...
3
votes
1answer
65 views
Infimum of the spectrum of an unbounded selfadjoint operator
Let $A$ be an unbounded selfadjouint operator in the Hilbert space $H$, having domain
$D(A)$.
Denoting by $\sigma_A$ the spectrum of $A$, we have
$\inf \sigma_A \ = \ \inf_{u\in D(A),\|u\|=1} \ ...
2
votes
1answer
46 views
Fixed-point method in many-dimensions
A well known method of easily solving multi-dimensional non-linear root finding problems, is to bring the equations into the form:
$$\bf x = g(x)$$
And then iterating. The problem is, one has to find ...
1
vote
0answers
42 views
Uniform Poincaré-Wirtinger constant for diffeomorphic domains?
Consider a bounded and connected open set (with regular boundary) $\Omega\subset\mathbb{R}^d$ and a family of $\mathscr{C}^1$-diffeomorphisms $(\varphi_t)_{t\in[0,\alpha]}$ all in ...
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0answers
31 views
Range of Self-Adjoint extensions
$\newcommand{\Rg}{\operatorname{Rg}}$
Let $S$ be a densely defined symmetric operator on a complex Hilbert space $H$, with defect index $n_{+}=\dim(\Rg(S+i)^{\perp})=n_{-}=\dim(\Rg(S-i)^\perp)=m < ...
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3answers
77 views
can I decide the following from the inverse of I-A?
I have a square matrix $A$.
Is it possible to determine if its largest eigenvalue is smaller (by magnitude) than 1 by inspecting the matrix $(I-A)^{-1}$?
(we can assume that $I-A$ is invertible.)
...
4
votes
1answer
157 views
Spectrum of this Operator
Let $A\colon \ell^{1}\to \ell^{1}$ be defined by $A(x)=(x_{2}+x_{3}+x_{4}+ \dots,x_1,x_2,x_3,\dots)$ where $x\in\ell^1$ iff $\sum|x_k|<\infty$.
Let $D$ be the closed unit disc in $\Bbb C$ and ...
1
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1answer
78 views
How can I calculate the eigenvalue of the following matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ such that its elements are all non-negative values.
I know that for any $k$, $A^k$ has elements on the diagonal which are smaller or equal to 1.
Can I ...
3
votes
1answer
74 views
is there a way to find or upper bound the largest eigenvalue of the following matrix?
I have a matrix $A \in \{0,1\}^{n \times n}$ -- i.e. a matrix with 1s and 0s only.
Is there a way to find or upper bound its largest eigenvalue?
I have a feeling it is related to connectivity of ...
0
votes
1answer
203 views
Finding Spectral Radius of Matrix
Find the Spectral Radius of $A=$
$\mbox{} \left[ \begin{array}{cc}
1 & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & -c & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} ...
4
votes
0answers
194 views
Spectral theorem for unitary operators
I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
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0answers
47 views
Process that is not harmonizable
Harmonizable processes include stationary time series, periodically correlated processes, and some transient time series. Zero-mean harmonizable processes also have an autocovariance function that ...
2
votes
1answer
169 views
spectral radius.
I am stuck in a problem of Conway's A course in a Functional Analysis. Can anyone give me a hint to solve the problem?
The question is "If $A$ is a Banach Algebra, then show that the function $r:A\to ...
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1answer
53 views
Proof that these Hessian matrix identities are similar matrices
I am wondering if $Q, P$ are similar matrices where for a function
$f:\mathbb{R^n}\to\mathbb{R}$ and for a diagonal matrix $D$
$Q=I-D^{-1}\nabla^2f(x)$ and $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$.
...
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1answer
152 views
Similar matrix proof
$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it ...
10
votes
1answer
259 views
The spectrum of an unbounded operator
It's well known that the spectrum of a bounded operator on a Banach space is a closed bounded set (and non-empty)on the complex plane. And it's also not hard to find unbounded operators which their ...
1
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1answer
91 views
spectrum of convex combination
$A,B$ are $n\times n$ Hermitian matrices. If the eigenvalues of $A$ and $B$ are all in an interval $I$, then the eigenvalues of any convex combination of $A,B$ are also in $I$.
In the book Bhatia, ...
2
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0answers
39 views
Strict positivity on the diagonal of a particular integral kernel: A question from Simon's Schrödinger Semigroups
This is a question pertaining to a (formerly?) open question from Barry Simon's Schrödinger Semigroups. In Theorem C.5.2 (page 504) of that publication, the existence of a specific function ...
1
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2answers
222 views
is it true that the infinity norm can be bounded using the $L_2$ norm the following way?
Let $v \in \mathbb{R}^k$, and let $A \in \mathbb{R}^{m \times k}$ and let $B \in \mathbb{R}^{m \times n}$ such that each column of $B$, $B_i$, has $$||B_i||_2 \le 1.$$
Is it true that:
$||v ...
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votes
1answer
72 views
Reading a DFT plot - did I get this right?
I am simulating the evolution of a liquid film through the solution of a 4th order nonlinear partial differential equations.
Of late, I began experimenting with DFT of the result that I have.
My ...
2
votes
0answers
47 views
Root Convergence rate of Iterative Scheme [closed]
I have an iterative sequence for optimizing an EM algorithm based loss function $L(X)$ with $t$ being the iteration number as:
$X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is a diagonal matrix, $B$ and ...
2
votes
1answer
150 views
Spectrum and orthogonal projection
Let $G$ be an operator on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$.
Let also $P$ be the orthogonal projection onto $\ker G$, $G_1$ the restriction of $G$ on $\ker P$ and
$G_2$ ...
3
votes
2answers
156 views
Inequality for a selfadjoint operator on Hilbert space
Let $T$ be a (possibly unbounded) selfadjoint nonnegative operator on a Hilbert space $H$
with domain $D$. Assume that $\langle T u, u \rangle \leq c$ for some $c>0$ and some $u\in D$.
I found ...
1
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0answers
43 views
When will the equality of Neumann's Trace inequality holds?
I am now studying the Neumann's trace inequality.
Some of the literature said that the equality holds when the two matrices have simultaneous ordered spectral decomposition.
Yet they don't really ...
11
votes
1answer
205 views
Singular-value inequalities
This is my question: Is the following statement true ?
Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators.
For every $n\in\mathbb{N}$ the following inequality holds:
...
8
votes
2answers
188 views
Are there n-th roots of differential operators?
In analogy to a Dirac operator, it seems to me that formally, the equation
$$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$
is solved by
$$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$
Is there a ...
1
vote
2answers
124 views
Spectrum of an element in sub-algebra: $\sigma_A(b)\setminus \{0\}\subseteq \sigma_B(b) \setminus \{0\}$
Please help me to prove this:(or give me some references for this.) Thanks very much!
Let $A$ be a (unital) algebra and $B\subset A$ a (unital) sub-algebra. Then for all $b\in B$: ...
0
votes
0answers
130 views
Diagonal Dominance and Spectral Radius
For positive semi-definite matrices, $A$ and $B$ with real entries,
Let:
$X=I-(2Diag(A)-B)^{-1}(A-B)$
The spectral radius $\rho(X) \leq ||X||$.
As, $(2 Diag(A)-B)$ becomes a better approximation ...
3
votes
1answer
189 views
are there any bounds on the eigenvalues of products of positive-semidefinite matrices?
I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$.
I am looking for upper bounds and lower bounds on the $m$-th largest eigenvalue of $AB$, in terms of the ...
5
votes
2answers
258 views
Why do we distinguish the continuous spectrum and the residual spectrum?
As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
0
votes
1answer
90 views
The row- and column-sums of a nonengative matrix with spectral radius less than $1$
Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$?
Edit: What if the matrix has positive ...
3
votes
1answer
110 views
Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign
I want to prove (or disprove) the following statement:
If $A$ is a square matrix with non-negative elements that has spectral
radius less then $1$, then any matrix obtained from $A$ by ...
4
votes
0answers
168 views
Continuous spectral value of right shift operator $\ell^2(\mathbb{N})$
Let $T:\ell^2(\mathbb{N} \to \ell^2(\mathbb{N})$ be the operator that sends$(x_1,x_2,x_3,...) \to (0,x_1,x_2,x_3,....)$. I want to show that $\lambda = 1$ is in the continuous spectrum.
To approach ...
5
votes
0answers
187 views
Why is the numerical range of an operator convex?
Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation}
It is a well-known fact that $W(T)$ is a convex subset of the complex ...
2
votes
1answer
118 views
Spectral radius and powers of a $2\times 2$ block matrix
The problem I'm struggling is the following:
Let $n$ be a positive integer and let $A=%
\begin{pmatrix}
B & C\\
C & B
\end{pmatrix}
\in\mathcal{M}_{2n}(\mathbb{R}_{+})$, where ...
1
vote
1answer
96 views
Spectrum Of The Laplacian Question
I'm given the following question (from Davies' book - Spectral theory of differential operators):
Use the theorem (*) below to prove that if $\Omega$ is a convex region in $ \mathbb{R}^2 $ , then:
$ ...
2
votes
2answers
124 views
Symbol Of Differential Operators
When we are given a differential operator of the form
$Lf : = \sum_\alpha a_{\alpha}(x) D^ \alpha f(x) $ , we can define the symbol associated with it to be the function:
$a(x,y) := \sum_\alpha ...
3
votes
0answers
78 views
Spectrum of a bounded linear operator [duplicate]
Possible Duplicate:
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $
Let $A$ be a bounded linear operator on a (complex) Hilbert space. I want to prove that ...
10
votes
1answer
283 views
Quantization of angular momentum: is Dirac's proof wrong?
I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
0
votes
1answer
80 views
Integral with spectral decomposition
Let $A:H\longrightarrow H$ be a self-adjoint operator, where H is an Hilbert space.
Let $(E_{\lambda})_{\lambda}$ be the spectral decomposition of $A$ and $\lambda_0$
a regular value of A with finite ...
6
votes
1answer
113 views
Do elliptic operators on Riemannian manifolds have a regularizing effect?
I'm working on my master thesis and need to handle some spectral theory of the Laplace operator on compact Riemannian manifolds and especially on the sphere. While investigating essential ...
