# Tagged Questions

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

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### 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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### 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 ...
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### Orthogonal projection onto the eigenspace of compact, self-adjoint operators.

Let $T$ be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that $f\in H$, $||f|| =1$ and $||(T-3)f||\leq 1/2$. Let P be the orthogonal projection onto the direct sum of all ...
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### Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it.
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### The topology of the spectrum of a linear operator

In general a spectrum of a linear operator has a decomposition into three parts: point spectrum, continuous spectrum and residual spectrum. What I'm interested in is the topology of these parts of ...
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### Clarification on point spectrum of an operator

From my understanding, if $\lambda$ is in the point spectrum, then $\lambda$ is a complex number such that it satisfies the equation $(T - \lambda I) x = 0$. My confusion arises from problems like ...
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### $\lambda$ is an eigenvalue iff spectral measure of $\lambda$ is nonzero

Let $M$ be a normal operator on a Hilbert space and let $E$ be the spectral measure of $\sigma(M)$ (the spectrum of $M$). Show that $\lambda$ is an eigenvalue to $M$ $\iff E(\{\lambda\})\not = 0$. ...
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### Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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### Spectrum of right shift operator in weighted $l2$ sequence space

Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
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### Laplace-Beltrami operator

I'm interesting in the Laplace-Beltrami operator on a sphere, more precisely its spectral properties including the spectral function, etc. So if someone can give me some references that treats this ...
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### Spectrum of nonnegative operator

Let $A$ be a bounded, nonnegative operator on a complex Hilbert space $H$. Prove that the spectrum $$\sigma(A)\subset[0,+\infty].$$ We say that an operator $A$ is nonnegative if it is self adjoint and ...
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### What theorem is this in the infinite dimensional case?

The spectral theorem for compact, self-adjoint operators is as I have understood the infinite dimensional case for orthogonal diagonalisation of a symmetric case in linear algebra? But in linear ...
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### Problem involving the Spectral Mapping theorem.

Consider the following problem: Let $T$ be a bounded operator in a Banach space $X$. Use the Spectral Mapping theorem to show that $|\lambda^n|\le\|T^n\|$ for all $\lambda\in\sigma(T).$ Here's ...
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I have the following two definitions in my notes: The spectrum of an operator: We define $\sigma(T)$, the spectrum of T, by, $$\sigma(T):=\{\lambda\in\mathbb C: T-\lambda I\,\, \text{is not ... 1answer 25 views ### Computing spectra in Banach algebras In general, computing the spectrum of a specific element in a Banach algebra can be very difficult. What are some of the less obvious tricks that you've encountered? 1answer 29 views ### How to find spectrum of a convolution operator Say k be s.t. \hat{k} is a bounded function on an LCA group G and Tf=f*k. Then T is bounded on L^2(G). Is there anything I can say about \sigma(T)? (except the properties that follow ... 0answers 22 views ### Holomorphic Functional calculus & approximate point spectrum Let A\in B\mathbb{(H)} and f\in Hol(A),then f({\sigma}_{ap}(A))={\sigma}_{ap}(f(A)). I only know how to prove f({\sigma}_{ap}(A))\subset{\sigma}_{ap}(f(A)),but have no idea on the other side. ... 2answers 96 views ### Consider the Banach Space C[0,1]. Find decomposition of spectrum of the indefinite integral operator. Cosider the Banach Space C[0,1] of real-valued continuous function on [0,1] with the supremum norm. and the linear operator$$A: x(t)\mapsto\int\limits_0^tx(s)ds$$Find its eigenvalues, ... 1answer 66 views ### Find the eigenvalues of the operator T. I have the following problem, "Suppose that X=\ell^1 and define the operator T\in B(X) as follows:$$Tx=\left(\frac12x_2,\frac13x_3,\frac14x_4,...\right)\,,\textit{where,}\,\,\, x=(x_1,x_2,x_3,.....
Consider the Hilbert space $\mathcal{H}=l^2(\mathbb{Z})$ and define the left shift operator $\mathcal{L}:\mathcal{H} \rightarrow \mathcal{H}$ by  \mathcal L (a_n) = (b_n) \qquad \text{ where } \...