# Tagged Questions

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

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Is anybody aware of an elementary proof that $T^*T$ is self-adjoint where $T$ is closed and densely-defined? All proofs I found so far use the Friedrich's extension or other more sophisticated ...
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### Random matrix on Clifford algebra within a specific grade

There has been some discussions about random matrices on generic Clifford algebra arXiv:1312.6291. However I would like to consider a more specific case by restricting the random matrix within a ...
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### Reference for the spectrum of the Bochner Laplacian on the 2-sphere.

I am looking for a reference for the spectrum of the Bochner Laplacian on $S^2$.
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### Detecting eigenvals of opposite sign

Consider a large (in the region of 500 by 500 up to 2000 by 2000) real, square symmetric matrix. What would be a good way of algorithmically determining if it has any pair of eigenvals with opposite ...
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### Convergence of spectrum [closed]

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
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### The Laplacian as the difference between mean value and a point

I am reading a book (Spectral Theory in Riemannian Geometry), which opens with an interpretation of the Laplacian. To get a better feeling about the Laplacian operator, consider for example a one-...
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### To prove this associative statement about spectral theory

Show that $\sigma(AB) \cup \{0\} = \sigma(BA) \cup \{0\}$ in general, and that $\sigma(AB) = \sigma(BA)$ if $A$ is bijective. I studied the associative statement of this somewhere but it did not ...
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### Convert partial fraction to continued fraction?

Lets say you have a partial fraction of the form: $$f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x}$$ Can anyone explain to me, in mildly plain English, how to convert this partial ...
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### Calabi's theorem

I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres). Theorem Let $\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$ be a full harmonic map. ...
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### Proof of Spectral Theorem

There are a number of results called Spectral Theorems. This question deals with the Linear Algebra result on normal operators, which has the self-adjoint case as a particular case. In class, we saw ...
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### Looking for a general approach to proving compactness of linear operators.

Preparing for my exam in functional analysis, I often have to prove that certain explicitly given operators are compact. I now have a decent amount of operators of which I can prove the compactness, ...
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### Using Fubini's Theorem in Contour Integrals proof

I have a few questions regarding the following proof: Suppose that $\mathcal{A}$ is a unital Banach algebra, and that $g$ is a complex-valued function which is analytic on $\sigma(a)$ while $f$ is a ...
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### Proof involving invertible elements of Banach algebra

I want to prove for a unital Banach algebra $\mathcal{A}$, it follows that if $\|a-b \| < \frac{1}{\|a^{-1} \|}$ then $b \in \mathcal{A}^{-1}$ (where $\mathcal{A}^{-1}$ is the subset of invertible ...
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### Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
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### Is this possible to decompose 2 arbitrary unitary matrices as: $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$?

I guess this statement is true, but I can't prove it: "For any 2 arbitrary unitary matrices $U_{1}$ and $U_{2}$, we can always decompose them into $U_{1}=ADB^{\dagger},U_{2}=BDA^{\dagger}$, where $A$ ...
The question is as follows: Let $T:X \to X$ be a compact linear operator on a normed space. If the $dim X= \infty$ then show that $0 \in \sigma(T)$. My attempt: Suppose on the contrary that $0 \... 1answer 27 views ### About finding the common diagonalizing similarity transformation. Say I have$2k$matrices$M_{a_1b_1}$,$M_{a_2b_2}$,..,$M_{a_kb_k}$and their negatives. Here$M_{a_ib_i}$is such that it has$0$everywhere except that it has$1$at$(a_i,b_i)$and$(b_i,a_i)$... 1answer 73 views ### Is the spectral radius of a matrix a convex norm of it? I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too. 0answers 38 views ### A question about minimizing the$\lambda_{max}$over a set of diagonal perturbations Say I have an off-diagonal symmetric$0,1,-1$entry matrix$B$and a set of$2k$diagonal matrices,$D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that$(1)$all the ... 0answers 34 views ### Given a spectrum, what can we know about its function? Say we are given a well-behaved function$f(t)$(either discrete or continuous) and are able to compute its spectrum$F(k)$using the (discrete) Fourier transform. Then say we lose$f(t)$and know ... 1answer 34 views ### significance and importance of spectral theorem I have started recently started Operator Theory and have been introduced to the Spectral Mapping Theorem: If$a \in \mathcal{A}$, where$\mathcal{A}$is a unital Banach Algebra and$f \in \text{hol}(a)...
I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ?...