Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
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Spectrum of two Hilbert spaces

Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and $sp_p(A) = sp_p(B)?...
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Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
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How to obtain $N_{\mu, i} (\lambda)=c_n \text{vol} (Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$? - Weyl's law

I am trying to prove the Weyl's asymptotic law for eigenvalues. In the document Weyl's law of p. $4$, I have managed to go up to the step $$\tilde{\nu_k} \leq \nu_k \leq \mu_k \leq \tilde{\mu}_k \...
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Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...
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$\forall f \in H^k$ is homogeneous of degree $k$, $f=r^k \hat{f}$, where $\hat{f}$ is in function of $\theta$ and $\phi$

In the document Spectral Geometry in Non-standard Domains at the end of page $39$, they display, without explanations, that $\forall f \in H^k$ is homogeneous of degree $k$, $f=r^k \hat{f}$, where $\...
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$A^{\frac 32}$ for $A\geq0$ self-adjoint as an integral of the Resolvent

Let $A\geq0$ be a bounded self-adjoint operator on a Hilbert space. I would like to show that $$A^{\frac 32} =c \int_0^\infty A^2 (y+A)^{-1}y^{-\frac 12}\text{d}y,$$ where $c>0$ is an appropriate ...
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Homogeneous polynomials - Explanation of $f(x,y,z) \in P^k$, then $f(x,y,z)= \sum_{i \geq 0} f_i(x,y) z^{k-i}$ [closed]

In the document Spectral Geometry in Non-standard Domains at the end of page $37$, they display, without explanations, that if $f(x,y,z) \in P^k$, then $f(x,y,z)= \sum_{i \geq 0} f_i(x,y) z^{k-i}$. I ...
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Find spectrum of integral operator

Let Af(x) = $\int_0^1 K(x,y)f(y)dy$, $A:L_2[0,1]\rightarrow L_2[0,1].$ Where $K(x,y) = \sinh(\min(x,y)\sinh(1-\max(x,y)). $ where $\sinh(x) = \frac{e^x - e^{-x}}{2}$ Find $\sigma(A), ||A||.$ I ...
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36 views

spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
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Weyl's asympotic law for eigenvalue : $\lim_{\lambda \to \infty} \frac{M(\lambda)}{\lambda}=\sum_p \frac{A(D_p)}{4 \pi}$

In the book Strauss W.A. Partial differential equations - An introduction (Wiley, $2008$, $1$nd Ed.) page $310 - 311$, it is probably a silly question, but is there anyone could give me a hint how to ...
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66 views

Find the spectrum of an operator

I am trying to learn some basic stuff about spectral theory, and I am a little bit lost. Please, could you help me and tell me how to find $\sigma(T)$ and $\sigma_p(T)$ of the operator $T:C([0,1]) \...
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29 views

Eigenvalues of block graph

Let us consider a graph $G$ having $m$ number of complete sub-graphs $K_{n_1},K_{n_2},...,K_{n_m}$ which have size $n_1,n_2,...,n_m$ respectively. Further $\forall i$, one vertex of $K_{n_i}$ is ...
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Courant-Hilbert's Book: Weyl's asymptotic law for eigenvalues - Planar domains

In the book Strauss W.A. Partial Differential Equations - an Introduction (Wiley, 2008, 1st Ed.) page $311$, there is a comment Now an arbitrary plane domain $D$ can be approximated by unions of ...
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37 views

Explanations : $\cup_{(j,k) \in E} S(j,k) ⊃E − (1, 1)$ $∩$ $($first quadrant$)$

I am stuck on a problem for a good while now. Is there anyone could tell me rigorously why $\cup_{(j,k) \in E} S(j,k) ⊃E − (1, 1) ∩ ($first quadrant$)$ of the problem of rectangle on page $18-19$ of ...
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29 views

If we know Spec($M_1$) and Spec($M_2$), what could we say about Spec($M_1 \cup M_2$)?

Let two domains $M_1$ and $M_2$ (Dirichlet conditions). If we know the spectrum of the Laplacian on $M_1$ and $M_2$, what could we say about Spect($M_1 \cup M_2$)? Is there a theorem that might give ...
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38 views

geometric interpretation of eigenfunctions of a vector field

Let $M$ be a smooth manifold and $X\in\mathfrak X(M)$ be a section on the tangent bundle. What is the geometrical interpretation of the eigenfunctions of $X$. That is functions in $f\in \mathcal C^\...
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The bigger the domain, the smaller the first eigenvalue - $\lambda(M_2) \leq \lambda(M_1)$ on the Laplacian

I know it is probably a silly question, but is there anyone could help me to complete of the corollary $3.1$ of that document? I pass a lot of time to try understanding the problem, but I can't ...
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62 views

How could we obtain $\lim_{n \to \infty} \frac{\lambda_n}{n}=\frac{4 \pi}{ab}$?

Related to the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$, is there anyone could explain to me how is it ...
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The point spectrum and residual spectrum of an operator on $l_2$ related to backward shift

I have a problem with the spectrum of this operator: $(Tx)_1 = x_2$ $(Tx)_2 = x_1$ $(Tx)_n = \frac{1}{n}x_{n+1}$ with $n\ge3$ Find the $||T||$, the point spectrum $\sigma_P(T)$ and $\sigma_P(T^{\...
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Weyl's asymptotic law for eigenvalue on the rectangle $D = \{0 < x < a, 0 < y < b \}$ - $N(\lambda) \geq \frac{\lambda ab}{4 \pi} - C \sqrt{\lambda}$

I have a few difficulties understanding the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$. I've managed to ...
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35 views

Square summable solutions of second order difference equations

There should be a standard argument for the following claim: If the system of equations of the form $$a_{n-1}u_{n-1}+(b_{n}-z)u_{n}+a_{n}u_{n+1}=0, \quad n\in\mathbb{Z},$$ where $a_{n}\in\mathbb{C}\...
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29 views

real spectrum clarification

Let $T: X\to X$ be a bounded operator on the real Banach space $X$.Does the spectrum of $T$ consist of the reals in the spectrum of its complexification?Or are they the same thing by definition?
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Weyl's asymptotic law for eigenvalues - Rectangle $D = \{0 < x < a, 0 < y < b \}$

Let the domain $D = \{0 < x < a, 0 < y < b \}$ in the plane. We now that $$\lambda_{n,m} = \frac{n^2 \pi^2}{a^2}+\frac{m^2 \pi^2}{b^2}$$ with the eigenfunction $$u_{n,m}= \sin(\frac{nπ}{a}...
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Dirichlet conditions - Proof of theorem $4$ on an example

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
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If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
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Dirichlet conditions - Explanation of the proof of theorem $4$

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
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If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
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How to find the Laplacian Eigen Values of the given graph

Find the Laplacian eigen-values of the of the graph on $N$ vertices whose edge set is given by $\{(i,i+1),1\le i<n\}$ and the edge $(1,n)$ . The answer is given to be $2-2\cos{\frac{2k\pi}{n}}$ . ...
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Max-Min Principle with an example on intervals

I have a few difficulties to understand the max-min principle (Intuitive understanding of Maximin Principle). Is there anyone could explain this theorem in using $[a,b] \subset [a',b']$? I know that ...
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Non square matrices and spectrum of $AB$

There is a known theorem (true in the general context of Banach algebras) that if $a,b \in A$ (where $A$ is a Banach algebra) then $\sigma(ab) \cup \{0\}=\sigma(ba) \cup \{0\}$ where $\sigma$ is the ...
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Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
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Intuitive understanding of Maximin Principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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Spectral theorem for matrices…

From the spectral theorem we know that if $A$ is a symmetric matrix then there exists an orthogonal matrix $M$ such that $A=M^{-1}DM=M^TAM$ My question is: if I have the matrix $A$ how do I find ...
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Distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi '>$ in term of Dirac-$\delta$-function

Personal question : Could it possible for the distribution distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi'>$ to be expressed in term of the Dirac-$\delta$-...
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Why are functional calculi interesting?

In my class on spectral theory we have defined the continuous functional calculus for normal elements of a C*-algebra. We were told that this is one of the most important results in spectral theory on ...
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finite spectrum eigenvalue

Let $T:X \to X$ be a linear bounded operator where X is Banach space ,and $\sigma (T)$ is a finite set.Then does the spectrum consist of eigenvalues only? Any hint or counterexample is ok. thanks in ...
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Strauss's book - Weyl's asymptotic law for eigenvalues

Someone told me that in the book Partial Differential Equations by Strauss you can find a proof of Weyl's asymptotic law for eigenvalues (one can hear the volume and dimension of a domain). Is there ...
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Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
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Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $...
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The reason of $\int_{-\infty}^{\infty}\mu_k^2(x)dx=1$

Is there anyone could tell me why if $$\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}=\int_{-\infty}^{\infty} (\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}} \mu_k^2(x))dx= \sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}...
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Compact resolvent proof

I want to prove the following proposition: We consider the operator $$A=A_V=-\Delta+\frac{1}{4}|\nabla V|^2-\frac{1}{2}\Delta V$$(where $V$ is a polynomial) with domain $D(A)=C_c^{\infty}(R^...
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36 views

Expansion theorem or Poisson Summation Formula? - Basis of eigenfunctions gives rise to a Fourier series

Does anyone could explain to me why in the Semiclassical's answer on the question Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula, the basis gives a series of the form $\...
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Set of normalized eigenfunctions - Expand some given function in the eigenfunction basis [closed]

The spectrum$(\mathbb{S}^1)=\{\lambda_k=k^2\ : k \in \mathbb{N}\}$, and the eigenfunctions $\mu_k(t)$ associated to the eigenvalues $\lambda_k$ are $a_k \cos kt + b_k \sin kt$ under the Laplacian ...
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130 views

Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^...
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How to prove $r(T_1 + T_2) \leq r(T_1) + r(T_2)$ when $T_1T_2 = T_2T_1$ for bounded linear operators?

Suppose $T_1$ and $T_2$ are bounded linear operators in a complex banach space and $r(A)$ is the spectral radius of $A$, satisfying $$ r(A) = \inf_{n>0} \|A^n\|^{1/n} = \lim_{n\rightarrow \infty} \...
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25 views

Coefficients of the eigenfunction

Related to the question : Eigenvalues of the circle over the Laplacian operator, how is it possible to find $c_1$ and $c_2$ related the explicit function $g(x)=c_1 \cos (\mu x)+ c_2 \sin (\mu x)$? ...
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1answer
59 views

Laplace-Beltrami operator on a circle - Explanation

I would like to find the spectrum of a circle. I know that the Laplace-Beltrami eigenvalue problem for unit circle is equivalent to the regular Laplacian eigenvalue problem for the interval of the ...
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1answer
35 views

The particularity of $k$ being an integer in the solution of a DE

Related to the Thomas's comment in the question : Eigenvalues of the circle over the Laplacian operator, is there anyone could tell me why the periodic function $g$ has a fundamental set of solutions ...
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44 views

Spectrum of linear operator, essential spectral radius

Consider the operator $L:L^1(S^1)\to L^1(S^1)$ given by $$ (Tf)(x)=\dfrac{1}{2}\left( f\left( \dfrac{x}{2}\mod 1\right)+f\left( \dfrac{x+1}{2} \mod 1 \right) \right) $$ where we identified $S^1$ with ...