# Tagged Questions

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

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### 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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### 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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### 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 ...