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

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Positive Map: Reduction

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. (Both possibly nonunital!) Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$ Implication: ...
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Show that $||(kI-T)^{-1}|| \le \frac{1}{d}$

Suppose that $T \in BL(H)$ where $H$ is a Hilbert Space. Let$k \in \mathbb{C}$. Let $d=distance(k,W(T)) \gt 0$. $W(T)=\{\lambda \in \mathbb{C}: \lambda=<Tx,x>, ||x||=1, x \in H\}$. Show that ...
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Expansion of an eigenfunction $\psi$ into a Fourier series

I was going through a paper by Jean Bourgain and it states that an eigenfunction $\psi$ of the Laplacian $\Delta$ with eigenvalue $-\lambda$ on the flat $n-$torus ...
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36 views

Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using, H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term ...
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23 views

irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
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21 views

Convergence criterion of vector fixed point iteration

As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$. For the vector variable version, it has been proved in the case when ...
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33 views

Maximal Accretive Implies Injective

I'm having trouble proving that $B $ is essentielly maximal accretive implies that there existes $ a>0 $ such that $(A^∗+a Id)$ is injective. where B is the closure of the operator A and A* is ...
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25 views

Equality of two measures

I am trying to prove the following statement. Let $F$ and $G$ be two finite measures on $((-\pi,\pi],\mathcal{B}((-\pi,\pi]))$ such that ...
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26 views

Fokker Planck operator is accretive

I would like to show that the Fokker Planck operator with domain $C_0^\infty \left(R^{2n}\right)$ defined by $$K= v.∂_x − (∂_xV (x)).∂_v + (−∂_v +v/2).(∂_v +v/2)$$ is essentially maximal accretive ...
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39 views

Uniqueness of positive square root of postive element in C* algebra

If a is a positive element then it has a unique positive square root, i.e. a unique b positive such that b^2=a. I understand the existence part of the proof. If ...
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44 views

Does the word “spectrum” in linear algebra have different meanings?

I'm reading several papers that refer to the spectrum as the set of all possible eigenvalues of a matrix, i.e., counting multiplicity, so that a list such as $\sigma = (\alpha_1, ... \alpha_n, 0, 0, ...
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32 views

Spectrum is finite or unbounded

Let $X$ be a Banach space and $A:D(A)\subset X\to X$ be a linear and closed operator with $\rho(A)\neq \emptyset$. Suppose that the map $$j:\left(D(A),\|\bullet\|_A\right)\hookrightarrow ...
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Determining eigenvalues of a differential or integral operator in Matlab?

Say I have a differential operator such as $L[\phi] = \frac{\partial \phi}{\partial x}$, or $L = \Delta \phi$, or an integral operator such as $L[\phi](x) = \int_{\partial D} \log(x - y) \phi(y) ...
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46 views

Spectrum of the derivative operator: What's wrong in my argument?

Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain ...
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63 views

Spectral Measures: Integrability

I really need this as tool for other threads! Given a Hilbert space $\mathcal{H}$. Also a Borel space $\Omega$. Consider a spectral measure: $$E:\mathcal{B}(\Omega)\to\mathcal{P}(\mathcal{H}):\quad ...
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Proving consequences of spectral decomposition of normal operator

$T$ is a normal operator on finite-dimension complex inner product space $V$. How do I use the spectral decomposition $T=\lambda_1T_1+\cdots+\lambda_kT_k$ to show: a) If $T^n=0$ for some ...
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Exponential of a self-adjoint operator

Let $\mathcal{H}$ be an Hilbert space. Firstly, I shall define some notions as their definitions may vary: A spectral resolution is a function $E:\mathbb{R}\to\mathcal{L}(\mathcal{H})$ (the space ...
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38 views

Under what conditions is the resolvent set of a linear operator connected?

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert Space, and assume that $T: H \to H$ is a possibly unbounded linear operator whose domain $D(T)$ is a dense subspace of $H$. As usual, we define ...
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26 views

Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
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42 views

Inverse spectrum problem - showing the existence of a 2x2 doubly stochastic matrix,

I am working through a couple of problems in Henryk Minc's book, Nonnegative Matrices, as a warm-up to understanding the inverse spectrum problem. This is Exercise 18 of Chapter VII of his book: ...
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81 views

An inverse spectrum problem in linear algebra,

I am reading the book, Nonnegative Matrices, by Henryk Minc, and came across an exercise that I would like to solve: Let $$\bar \sigma = (\bar\lambda_1, ... , \bar \lambda_n)=(\lambda_1, ... ...
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51 views

Spectral radius as an upper bound to norm

The operator norm of a matrix is an upper bound for the spectral radius, and equality holds in particular when the matrix is Hermitian. One proof of this uses the spectral theorem. Is there a direct ...
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43 views

Proof compactness of adjoint operators

I am trying to understand a proof of the following statement: Given a complex B-space X and a compact operator $T:X\rightarrow X$, the adjoint operator $T^\ast:X^\ast \rightarrow X^\ast$ is compact as ...
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157 views

Fact regarding Kirchhoff's Theorem

Question regarding Kirchhoff's Theorem: If $ L(G)$ denotes the Laplacian of a graph $G$ then Kirchhoff's Theorem states that number of spanning trees in $G$ is equal to $(-1)^{i+j} \det L(i|j)$ ...
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36 views

Matrix -tree theorem-How to understand the theorem

I am having trouble understanding Kirchhoff's Theorem. The statement I want to prove is that if $\lambda_1,\lambda_2,...,\lambda _{n-1}$ are non-zero eigen values of $L(G)$ then Number of ...
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21 views

Inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$

I want to calculate the inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$ where, $w \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector and $ \alpha $ is some ...
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29 views

Compact operators form the only closed proper ideal of bounded linear operators

I am trying to understand the following proof in Trace Ideals and Their Applications by Barry Simon (Proposition 2.1): Let $\mathcal{J}$ be a two-sided ideal in $\mathcal{L}(\mathcal{H})$ containing ...
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93 views

Can $e^{ax}$ be said to be the eigenfunction of the operator $\frac{d^{(n)}}{dx}$?

I'm gradually getting familiar with operators (as they are used in QM) and the terminology surrounding them, and I was wondering whether all the (to me) well-known operators have straight-forward, ...
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What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where ...
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Image density in spectral theory

The operator $T$ is $\dfrac{d}{dt}$ and $$\left\{\begin{array}{lc}x'(t)-\lambda x(t)=-y(t)\\ x(0)=0\end{array}\right.$$ and the domain of $T$ is $D(T)=\{x\in L^2(0,\infty):\; x\; \text{absolutely ...
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Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
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Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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What do Fourier Series for Other Symmetric Operators Look Like?

I understand that Fourier analysis works (up to constant multiples) by considering the inner-product space $E$ of smooth functions $[-\pi,\pi] \to \mathbb C$ with inner product. . . $\displaystyle ...
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Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
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Spectral theorem for unitary operators T o F

Si $T$ es unitaria y $B$ es una base de $V$ formada por vectores propios de $T$ entonces $B$ es un conjunto ortogonal. If $T$ is unitary and $B$ is a basis for $V$ consisting of eigenvectors of $T$ ...
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Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
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Uniqueness of spectral decomposition

Suppose $T: V\rightarrow V$ is diagonalizable on an arbitrary vector space (not necessarily an inner product space), so $T = \sum_{i=1}^r\lambda_i P_{\lambda_i}$ where ...
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Self-adjoint operators, projections, and resolutions of the identity.

In my Functional Analysis course, we're discussing the Spectral Theorem and the like. One question from a previous exam states the following: Let $H$ be Hilbert over $\mathbb C$, let $T \in B(H)$ be ...
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Example of operator with spectrum equal to $\mathbb{C}$?

In my Functional Analysis course, we proved that for a (possibly unbounded) operator $T$ that is densely defined, closed, and symmetric, exactly one of the following four occurs: $\sigma(T) = ...
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Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
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28 views

Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator ...
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Spectrum of operator $T((x_n)_{n\in\mathbb{Z}})=\left(\frac{1}{n^2+1}(x_n-x_{-n})\right)_{n\in\mathbb{Z}}$

The eigenvalues should satisfy: $$T(x_n)=\lambda x_n$$ $$\frac{1}{n^2+1}(x_n-x_{-n})=\lambda x_n$$ $$\left[(n^2+1)\lambda+1\right]x_n=x_{-n}$$ I suppose that this should mean that ...
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Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually I'm not exactly looking into bipartite but the ...
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Can one hear the *material* of a drumhead?

"Can one hear the shape of a drum?" is a well known problem, originating from Kac, 1966, that questions whether an (idealized) drum head is completely specified by its spectrum. That is: is the ...
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63 views

Spectrum of operator in $l_2$

Sorry guys, I have a problem with this exercise. Let T an operator in $l_2$ Hilbert space: $$ (\textrm{T}x)_1 = x_2 , $$ $$ (\textrm{T}x)_2 = 0 , $$ $$ (\textrm{T}x)_n = x_{n-1} - x_n ...
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How to prove that $f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda$ where $f_{\lambda} $ is the eigenfunctions of $\Delta$

On Euclidean space $\mathbb R^n$, how to prove that $$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$ where $\Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) $, whith $\Delta$ is the ...
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References on the spectral theory of the Floquet problem

The Floquet problem is about the linear ordinary differential equation $$ \dot{\psi} = A(t) \psi. $$ Here $A(t)= A(t+T)$ is a periodic $n\times n $ matrix. Suppose $A(t) =-i H(t)$ with $H(t)$ being ...
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Proof: the resolvent operator is holomorphic.

I tried to prove that the resolvent operator $$\rho(A) \to \mathbb C,\space \lambda \mapsto R_{\lambda}(A):=(\lambda id_X -A)^{-1} $$ is holomorphic, where here $A$ is a bounded linear operator from ...
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Nonempty intersection between approximate point spectrum and residual spectrum

On the Wikipedia page on "Spectrum (functional analysis)", it is mentioned that the approximate point spectrum and residual spectrum are not necessarily disjoint. Is there a straightforward example to ...
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35 views

Different versions of Mercer's theorem

I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem: About the positive-definiteness conditions. In the Wikipedia pages on RKHS ...