# Tagged Questions

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

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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 n, ...
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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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### 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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### 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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### 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 ...
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 ...
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 ...