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

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If a normal element of a C* algebra has real spectrum, then it is self-adjoint

Let $A$ be a $C^*\!$-algebra. Suppose $x$ is a normal element of $A$ and $\operatorname{spect}(x)$ lies in $\mathbb{R}$. Prove that $x$ is self-adjoint. I tried the following: using ...
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20 views

Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
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47 views

Difference between continuous and essential spectrum, examples?

Can anyone help me to understand the definitions of the continuous and essential spectra in simple terms and point out the difference on examples where the two definitions coincide and where don't? ...
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Thresholding in spectra of partial traces of random symmetric matrices

I found an interesting behavior while looking at partial traces of random matrices. This is something I was studying numerically, and I haven't completely ruled out the possibility of numerical ...
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46 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
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Solving for the spectrum and eigenvectors of the “shift operator(?)” $T$ in $P_3(\mathbb{R})$?

This question is inspired from accidentally misreading this question in my tutorial exercise in my linear algebra course because I forgot the 3 in $P_3(\mathbb{R})$ (The actual question can be ...
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Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
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33 views

Spectrum of integration operator on $C[0,1]$.

I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by: $$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$ I can show that $0$ is contained in the approximate point spectrum with ...
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67 views

Central Limit Theorem proof: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
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36 views

What do we know about inverses of matrices which are “like” Laplacians of graphs?

Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)] You can consider variations of $L$ like ...
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34 views

Spectrum of double infinite shift using isometry to Fourier series

I'm trying to find the spectrum of the operator $T: l^2(\mathbb{Z}) \to l^2(\mathbb{Z})$ given by right shift but I am having some difficulties. I can show that $l^2$ is isomorphic to ...
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Interesting examples of non-normal operators?

I am currently learning spectral aspects of linear algebra. At first sight, it seems like normality is very narrow restriction. But, I can not think up any examples of non-normal operators. There is ...
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17 views

Cauchy integral type formula for self-adjoint operator

Let $\Gamma$ be a differentiable Jordan curve in the resolvent set $\rho(A)$ of the self-adjoint operator $A$. How does one show $\chi_\Omega(A) = \int_\Gamma R_A(z) dz$, where $\Omega = Int \Gamma ...
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27 views

How networks with high largest eigenvalues are more robust?

In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to ...
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22 views

Method for ?not quite? weighted least squares fitting for more realistic results

I need a linear least squares type of fitting algorithm that understands how to weight the probability of a response coming from certain functions over another. To explain, given the standard linear ...
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28 views

Commutative multiplier algebra

In my course of spectral theory and operator algebras I came across the following exercise: Let $\mathcal{A}=C_0(X)$ where $X$ is a locally compact Hausdorff space. Describe the multiplier algebra ...
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70 views

Is there any relation between the Gershgorin circles of a matrix and its resolvent?

Let $A$ be a real symmetric matrix. Now fix a diagonal index say "i" and let $x > max-eigenvalue(A)$. Now is there any thing known about the Gershgorin circle of $[1/(x-A)]_{ii}$ in terms of the ...
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A linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an orthonormal basis.

I am studying for a final exam and came across a sentence in my linear algebra textbook stating that "a linear operator T is normal if and only if there $[T]_\beta$ is normal, where $\beta$ is an ...
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38 views

Spectra of operator matrices

Suppose we are given a bounded linear operator $A\colon X\to X$ on a Banach space which is injective and has closed range. Can we find two other operators $T$ and $S$ say such that ...
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24 views

The spectral projections of convolution operator

Given a self-adjoint operator $A$ in a Hilbert space $H$. How can one find its spectral projections $\{E_{\lambda}\}_{\lambda\in\sigma(A)}$? In particular, given a convolution operator on $L^2(G)$, ...
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21 views

Convergence of powers of products with diagonal matrices

Suppose that $M$ is an $n\times n$ matrix with $\rho(M)<1$ (i.e. its maximum absolute eigenvalue is less than $1$). Is the following statement then true? If $\forall t\in\mathbb{N}$, $D_t$ is ...
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Spectral theorem for compact normal operators

Let $H$ be a Hilbert space and $A$ a compact normal operator from $H$ to $H$. How to show that its eigenspaces produce the space? I can show it for self-adjoint operators and by setting ...
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1answer
51 views

Spectral theory for $f\mapsto f\circ g$

Consider the Banach space $B = C([0,1] \to \mathbb R)$ of continuous functions from $[0,1] \to \mathbb R$ with the supremum norm. Let $g$ be a continuous function $g:[0,1] \to [0,1]$. Then one can ...
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Algebra with element having empty spectrum?

The definition of the spectrum makes sense for any algebra. I guess we can go to the unitization to make sense of it even non-unital algebras. Recalling the well-known fact that for normed algebras, ...
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15 views

Show that an operator is closable

Let $H=\mathcal{L}^2(\mathbb R^2,dxdy)$ and let $A$ the operator defined by: $$ A[f](x,y)=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+i(y\frac{\partial f}{\partial ...
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Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
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27 views

Seminar concearning Spectral Theory of Differential Operators?

I must prepare a seminar about spectral theory of linear partial differential operators. However, I'm at a loss as to a nice reference. I'm looking for something that fits in a graduate spectral ...
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18 views

Eigenspaces and eigenfunctions of the multiplication operator

Let me consider the Borel functional calculus. Denote by $A_{\varphi}$ the operator of the multiplication by a bounded borel measurable function $\varphi$, i.e. $$ L^2(X)\ni f \mapsto ...
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Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
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171 views

What is the importance of phase spectrum in Fourier transform

For any given signal using Fourier transform, we can compute it's magnitude and phase spectrum. But I have found that while discussing Fourier transform ,only frequency spectrum or magnitude ...
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32 views

Question about the spectrum of linear (unbounded) operator

I'm not much confident with functional analysis, but I found in my lecture note a statement that doesn't convince me. For a linear (possibly unbounded) operator $T$ in a Banach space the following ...
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Positive operators acting on a sequence of vectors

Let $A$ be self-adjoint, unbounded operator with domain $\mathcal{D}\subset \mathcal{H}$ ($\mathcal{H}$ - Hilbert space). We assume that the spectrum of $A$ is absolutely continuous and is the set ...
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A relation between the domain of $A$ and the domain of $\bar A$

Let $A$ be an operator: $$ A:D(A)\to R(A) $$ where $D(A)$ and $R(A)$ are respectively the domain and the range of $A$ and they are subspaces of a Hilbert spcae $(H,\|\|)$. Suppose that $A$ is a ...
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33 views

Borel functional calculus and multiplication operator

Let $A_f$ be the multiplication operator in $L^2(\mathbb R)$ with the function $f$. If $g$ is a bounded Borel function on $\mathbb R$, why is $g(A_f)$ defined by the functional calculus the ...
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30 views

Projection valued measure of bounded self-adjoint operator.

Let $A$ be a bounded self-adjoint operator with $P_E=\chi_E(A)$ as its projection valued measure on set $E\subset \mathbb{R}$, then $f(A)=\int f(\lambda)dP_\lambda$ and $A=\int \lambda dP_\lambda$. ...
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Self adjoint operator property

Let $A$ and $B$ be two self adjoint operators on $L^2(\mathbb{R}, \mu)$ and $L^2(\mathbb{R}, \gamma)$, suppose the spectral measure $\mu, \gamma$ are absolutely continuous. Show that $A$ and $B$ are ...
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spectral theory of Laplacian on $\mathbb R^n$ [duplicate]

Can you describe the spectrum of the Laplacian $ \Delta : H^2(\mathbb R^n) \subset L^2(\mathbb R^n) \rightarrow L^2(\mathbb R^n)$? I am interested for which values $z \in \mathbb C$ the equation ...
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35 views

Which operators other than self-adjoint operators have no purely imaginary eigenvalues?

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
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stationary process with discontinuous spectral distribution function

Let's say we have a zero mean stationary process $X_t$ with spectral distribution function $F$, then the autocovariance function of $X_t$ can be written as ...
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1answer
30 views

Point spectrum of operator on $\ell^2$?

Considere the bounded linear operator $S:\ell^2\longrightarrow \ell^2$ given by $$ S(\xi_j)_j:=\left(\frac{\xi_2}{1}, \frac{\xi_3}{2}, \frac{\xi_4}{3}, \ldots\right).$$ How to show the point spectrum ...
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Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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20 views

Spectrum of adjoint operators

If $A \subset \mathbb C$, we set $A^* = \{\bar z: z \in A\}$. I want to prove the following theorems. $\rho(T)^* = \rho(T^*)$ and $\sigma(T)^* = \sigma(T^*)$. $\sigma_c(T)^* = \sigma_c(T^*)$. ...
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Two definitions of spectrums

In Kreyszig's Introductory Functional Analysis Page 371, the point spectrum is defined as $\sigma_p(T)$ such that $R_\lambda(T) = (T - \lambda I)^{-1}$ does not exist. While in my functional ...
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51 views

A continuous field of C* algebra, $C(\mathbb T)\rtimes\mathbb Z_2$

Given a $C^*$-algebra, $A=${$f:[0,1]\rightarrow M_2(\mathbb C)$ where $f(0),f(1) $ are diagonal } which is isomorphic to $C(\mathbb T)\rtimes\mathbb Z_2$, How can I determine its continuous field ...
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Eigen value and Regular Graph (not Strongly Regular graph).

$A,B$ are 2 adjacency matrices of $d$ Regular graphs(not Strongly Regular graphs). I would like to know- 1.Results/ information related to Eigen values of A,B. There is a formula for ...
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64 views

A property of measurable functional calculus

A seemingly simple property of the measurable functional calculus: Let $A$ be a self-adjoint operator on a Hilbert space $H$ and let $P$ be the associated projection-valued measure, such that $A = ...
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1answer
39 views

Evaluate the spectrum of a bounded linear operator

$H$ is a separable Hilbert space over $\mathbb C$ and $\{u_n\}$ is a maximal orthonormal set of H. $A \in B(H)$ and there exists $\lambda \in \mathbb C$ such that $$A(u_n) = \lambda u_n - u_{n+1}, n = ...
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1answer
73 views

Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
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18 views

Spectrum of multiplication operator and point masses

Consider the multiplication operator $A$ on $L^2(\mathbb R,d\mu)$, $Af = \lambda f$ for some function $\lambda \in L^2$. Then $f \in L^2$ is in $\ker(A-z)$ (for some $z \in \mathbb R$) implies that ...
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63 views

Spectral measure associated to eigenvector of self-adjoint operator

Let $A$ be a self-adjoint operator on .the Hilbertspace $H$ and let $\lambda_0$ be an eigenvalue of $A$ with corresponding eigenvector $\psi$. The spectral theorem tells us,that there is a ...