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

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Selfadjoint Operator: Basic Criterion

For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way?
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Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
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What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
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50 views

Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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Is this some kind of Perron-Frobenius theorem?

If one has a matrix $A$ whose entries are all $\geq 0$ (or $>0$) then there apparently exists a diagonal matrix $D > 0$ (i.e all the diagonal entries are positive) and a constant $\alpha >0$ ...
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For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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51 views

Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...
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43 views

Weyl's law, meaning of the asymptotic formula, does it imply a bound?

Weyl's law states the eigenvalues of the Laplacian behave as $$\lambda_j \sim f(n)j^{\frac 2n}\quad\text{as $j \to \infty$}$$ where $n$ is the dimension. Does this literally mean that, $$\lim_{j \to ...
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Self-adjoint operator- domain unique?

I was wondering about the following: Let $T : dom(T) \subset H \rightarrow H$ be a self-adjoint operator, does this mean that the domain of $T$ is uniquely defined or is it possible to make the same ...
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interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
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17 views

Faber-Krahn inequality for domain in Z^d with nearest-neighbor connections

In $\mathbb{R}^d$ there is a theorem that if you are looking for the first Dirichlet eigenvalue $\lambda_1$ of a domain $D \subset \mathbb{R}^d$ with a given volume $V$, then $\lambda_1$ will be ...
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60 views

Spectrum of convolution operator

I was trying to find the spectrum of the convolution operator $$ J \ast u = \int_D J(x-y) u(y) dy $$ for bounded domain $D \subset \mathbb{R}$. Does anybody know it or have a reference for me? ...
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Positive Elements: Norm (Decomposition)

Given a C*-algebra $\mathcal{A}$. Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$ Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$ ...
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Modifications to the definition of graph Laplacian?

Many people have defined various definitions for graph Laplacian. For example see here [1]. What is common between various definitions of Laplacian that makes all of them ``Laplacian"? For example ...
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Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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Eigenvalues of compact operator don't have nonzero accumulation points

In the book Elements of the theory of functions and functional analysis of Kolmogorov and Fomin, there is a proof of the following theorem, Every compact operator $A$ on a Banach space $E$ ...
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Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
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Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...
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Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
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866 views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
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Invariant subspace of bounded self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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Prove of some properties about unitary operators [closed]

Let $X$ be a hilbert space and $T\in L(X)$ be an unitary operator. Show (1) $\sigma(T)\subset\{\lambda \in \mathbb C:|\lambda|=1\}$ (2) for $\lambda \in \mathbb C$ with $|\lambda|\neq1$ holds: ...
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50 views

Invariant subspace of self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
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207 views

Every normal operator on a separable Hilbert space has a square root that commutes with it

Show that every normal operator on a separable Hilbert space has a square root that commutes with it. Uniqueness? My attempt: Let $T$ be a normal operator. By polar decomposition $T=U|T|$ where ...
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38 views

Image of a projection

Show that $\lambda$ is an eigenvalue for normal bounded linear operator $N$ on Hilbert space $H$ with spectral measure $E$ iff $E(\{\lambda\})\neq 0$, in which case the range of $E(\{\lambda\})$ is ...
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1answer
26 views

Finding an explicit formula for $\int_\Gamma (z - A)^{-1} dz$, where $A$ is an operator on $l^p$

I am working on an example from the spectral theory of linear operators on Banach spaces. This is example 6.10 from chapter VII of Conway's A Course in Functional Analysis. Let $\{\alpha_n ...
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1answer
64 views

Self adjoint operator has spectrum around 0 and 1

I need to prove the following statement: Let $\mathcal{A}$ be a unital $C^*-$Algebra, $A$ a self-adjoint element and $P$ a projection, so $P^2=P=P^*$. Let $\delta :=\|P-A\|$. I want to prove that ...
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Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...
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Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
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Dimension of generalized eigenspace.

Let $T \in \text{Hom}_F(V,V)$, suppose the characteristic polynomial of $T$, $c_T(x) = (x- \lambda)^kp(x)$, where $p(\lambda) \neq 0$, show that $\text{dim}_F (E_{\lambda}^\infty) = k$, where ...
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36 views

Special Elements: Spectrum

Given a C*-algebra with unit $1\in\mathcal{A}$. For normal elements one has: $$A^*=A^{-1}\iff\sigma(A)\subseteq\mathbb{S}$$ $$A^*=A\iff\sigma(A)\subseteq\mathbb{R}$$ ...
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Distance random matrix

In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by ...
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Construct a bounded linear operator S on H such that σ(S) = A

Given an infinite dimensional Hilbert space $H$. Let $A\subseteq \mathbb{C}$ be closed and bounded. Construct a bounded linear operator $S$ on $H$ such that $\sigma(S)=A$, where $\sigma(S)$ is the ...
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65 views

Positive Operator: Norm Estimate

In class we encountered the statement: $$H\geq C1\quad(C>0)\implies\|\mathrm{e}^{-\beta H}\|<1\quad(\beta>0)$$ How does one prove this? Moreover, what about the weakened version: $$H\geq ...
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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 ...
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116 views

Positive Elements: Characterization

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Define positive elements as: $$A\geq0\iff\sigma(A)\geq0\quad(A=A^*)$$ Positive elements can be characterized by: $$A\geq0\iff A=B^*B$$ ...
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Symmetric Operator vs. Real Spectrum

For symmetric operators one has a characterization: $$A\text{ symmetric}:\quad A=A^*\iff\sigma(A)\subseteq\mathbb{R}$$ (I want to investigate to what extend symmetry is a necessary assumption.) ...
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spectrum of the right shift operator

Here is the question: Considering the right shift operator $S$ on $l^2({\bf Z})$, what can one know about ran$(S-\lambda)$? Here is what I thought: If one wants to prove that the operator ...
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Shifting of the spectrum of a linear operator - in both the symmetric and non-symmetric cases,

a) I finished a problem that sort of highlighted the fact that if a real symmetric matrix $A_2$ = A + I, where A is also real and symmetric, then $A_2$ has the same eigenvectors as A, but its spectrum ...
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Image of a commutative C*-algebra

Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism ...
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Spectrum of left shift operator: take two

This is my second attempt at calculating the spectrum of the left shift operator on a Hilbert space. I got stuck again and I would be grateful if someone could help. (You can find my previous (failed) ...
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Polynomial Calculus: well defined?

Disclaimer This thread has been refreshed! Problem Given a C*-algebra $\mathcal{A}$. Consider an element $A\in\mathcal{A}$. Introduce an abstract polynomial calculus: $$A=A^*:\quad ...
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Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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Polynomial Ring: Root vs. Remainder

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
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Symbol of an inverse on an invariant subspace of a certain elliptic operator

I have little experiences in dealing with pseudodifferential operators, so I hope I could get some hints for the following question: Let $L$ be an 2nd-order elliptic operator on $\mathbb{T}$ with ...
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Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...
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Spectrum of left shift operator $L\in B(H)$

Let $H$ be a Hilbert space with an orthonormal base $e_i$ and $L$ the left shift operator $L\in B(H)$: $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$. I computed the spectrum could someone please tell ...