# Tagged Questions

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

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### Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
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### Construction of a Strongly Regular Graph which has regular Neighbourhood graphs in all iteration.

Notation and Definition: $G$ is a Strongly Regular Graph (not complete or a cycle) and is denoted by $\mathrm{SRG}(n,r, \lambda, \mu)$ if it has the following properties: Every two adjacent ...
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### Trace evaluation via complex analysis

We are given $U$, $V$ unitary matrices of size $N \times N$ whose spectral decomposition is known (in my specific problem, $N=4$, and $U$, $V$ are matrices with real coefficients but we can keep it ...
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### Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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### How can I tell that my matrix is nilpotent?

I just computed a 15x15 matrix by hand :( It is not upper triangular as I hoped it would be. But my computations agree with what's offered in the student solution. My question is: the solution ...
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### spectral measure and normal operators range

Let $N$ be a normal operator with spectral measure $E$. We want to show that if $N=\int z\ dE(z)$ and $ε>0$, then $\operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N$. Is this true? Let ...
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### Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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### normal operators spectral decomposition

Let $N$ be a normal operator with spectral measure $E$. We want to show that if $N=\int z\ dE(z)$ and $ε>0$, then $\operatorname{ran} E(\{z∶ |z|>ε\})⊆\operatorname{ran}N$.
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### Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
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### what is a spectral function?

My knowledge in spectral theory is very limited, but lately I heard talking about the spectral function of an operator and how it's important. By curiosity I tried to look for a definition and a ...
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### Solving for the spectrum and eigenvectors of the “shift operator(?)” $T$ in $P_3(\mathbb{R})$?

This question is inspired from accidentally misreading this question in my tutorial exercise in my linear algebra course because I forgot the 3 in $P_3(\mathbb{R})$ (The actual question can be ...
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### Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
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### Spectrum of integration operator on $C[0,1]$.

I'm trying to find the spectrum of the operator $T: C[0,1] \to C[0,1]$ given by: $$T(f)(t) = f(0) + \int_0 ^{t} f(s) ds$$ I can show that $0$ is contained in the approximate point spectrum with ...
Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)] You can consider variations of $L$ like ...