Spectral theory is a study of generalized notions of eigenvalues and eigenvectors.

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Spectrum of perturbed operator's

Let $G$ be a normal operator with compact resolvent acting on a Hilbert space $H$ such that $\ker G \neq \{0\}$. If $P$ denotes the orthogonal projection onto $\ker G$, and if $\{\lambda_n\}$ are ...
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If $\lambda$ is isolated in $\sigma(u)$, then $E(\left\{\lambda\right\})(H)=\ker(u-\lambda)$.

This is a Question 2.11 from Murphy's book: C$^*$-algebras and Operator Theory: Let $H$ be a Hilbert space. Let $u\in B(H)$ be a normal operator with spectral resolution of the identity $E$. (a) ...
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About some properties of the heat kernel

Maybe my questions are stupit, but I'm so unsure about some properties of the heat equation. Suppose we have a compact and smooth manifold M (possibly with boundary), then in a lot of books they ...
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characterisation up to unitary equivalence

My book says that the spectral theorem for compact normal operators characterises compact normal operators up to unitary equivalence. It doesn't expand on this so I was wondering what does this mean ...
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The eigenspace of 0 problem

In the spectral theorem for a compact normal operator, do we exclude the eigenspace corresponding to 0 (assuming its an eigenvalue) from the space decomposition. My reason for asking is this: Every ...
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49 views

properties of norms of operators

Let $T$ be a linear densely defined operator on a Hilbert space $H$ and $L$ be a selfadjoint operator with discrete spectrum such that $L^{-1}$ is bounded and $$\|Tf\| \leq \Phi_{\eta}(f) + \eta ...
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Spectrum of a compact operator

If the spectrum of a compact operator is finite, I don't understand why $0$ has to be a member. I have proved that for all $\epsilon > 0$, there is only a finite number of eigenvectors which have ...
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121 views

Direct sum of eigenspaces of a compact operator has finite codimension

In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks.
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operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
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64 views

Spectral decomposition

For a compact normal operator, the space can be written as the sum of generalized eigenspaces. So every element can be written as a linear combination of the eigenvectors, one from each eigenspace. ...
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62 views

Spectral decomposition for normal compact operator

My book says $Tx=(\alpha x_{\alpha})$ where the $\alpha$ are eigenvalues the of T. But the image of an operator is not in general a sequence. Do they mean these are the scalars in the linear ...
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pierre simon laplace and his knowledge of the (Laplacian) matrices

so as we all know, there is a graph matrix called the Laplacian that is used in some eigenvalue/eigenvector/graph theory/spectral theory problems. i'm wondering if the name of this matrix is ...
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Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
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Clarification about Arveson exercise

In an exercise at the start of the book of Arveson "A short course on Spectral Theory"(page 5 exercise 5), while asking to prove that the operator $Kf(x)=\int_{0}^{1}k(x,y)f(y)dy$, (where k is a ...
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104 views

Projections in spectral decomposition.

In my quantum mechanics book the spectral decomposition of operator $A$ is given as $A=\sum\limits_j\lambda_jP_j$ where $\lambda_j$ are the eigenvalues of matrix $A$ and $P_j$ is the orthogonal ...
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48 views

Resolvent's estimation.

Let $H$ be a Hilbert space and $L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I would ask about this inequality because I don't understand it$$\displaystyle{\|(L- \lambda ...
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Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...
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52 views

Perturbation of a matrix with negative eigenvalues

Let $A$ be a square matrix with all eigenvalues negative. What is the relationship between the $\lambda_\max$ of perturbed matrix $A + X$ and the norm of the perturbation $\|X\|$? PS: I know that the ...
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92 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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82 views

Fast Gauss-Seidel convergence on low rank matrices

I stumbled upon the following remarkable fact when experimenting with the Gauss-Seidel iterative solver: First I construct a low-rank symmetric positive semi-definite matrix $A = M^TM$ with M a ...
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How to show that the spectral radius of a binary tree approaches exp(1) as the N tends to infinity?

How can I prove mathematically that the spectral radius of a binary tree approaches e as the number of nodes tends to infinity? I mean it is true that the increase in nodes number is exponential but ...
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54 views

Characterization of all matrices with unit spectral radius under constraint

Let $A \in \mathbb{R}^{n \times n}_{\geq 0}$ be a symmetric matrix with positive row sums $\mathbf{d} := A\mathbf{1} > 0$. I am interested in characterizing all those positive diagonal matrices $Z ...
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Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
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Spectrum of product from two selfadjoint matricies

If I have 2 selfadjoint matricies A,B given, is the spectrum of $A B$ always real? I know that $A B$ is not necessary a selfadjoint matrix, but are some properties of the spectrum preserved?
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A transformation recipe for functional calculus of a self-adjoint operator?

Consider a self-adjoint operator $\operatorname{A}$ on a Hilbertspace $\mathcal{A}$ and its spectral decomposition according to the spectral theorem: $$A = \int_{\mathbb{R}} \lambda \;dP_\lambda$$ ...
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Spectrum of the Orr Sommerfeld equation

The Orr Sommerfeld equation is as follows $$\psi''-k^2 \psi - \frac{U''}{U-c}\psi=0$$ where $\psi(y)$ is a complex valued function on $[0,2\pi]$ satisfying Dirichlet boundary conditions ...
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198 views

Resolvent R(1) of the Laplace operator not compact

I want to show that $$R_\Delta(1):=(1-\Delta)^{-1} $$ is not compact in $\mathbb{R}^3$. I have found that for $\chi_{B}$ being the characteristic function for a set $B\subset\mathbb{R}^n$, ...
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331 views

Spectrum of operator

Like my previous question, I'm considering the same space and operator: Hilbertspace adjoint But this time I am trying to determine the spectrum of $T$. I feel like I'm messing up my definitions a ...
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69 views

invariant inner product on eigenspace

I have several questions about the following corollary: "Let G/H be a riemannian homogeneous space where G is a compact Lie group. Let $E_{\lambda}=\lbrace f\in C^{\infty}(G/H) : -\Delta f= \lambda ...
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regarding a proof of $\|\theta(e^{i\lambda})\|^2$

When studying the spectral representation of time series, I read the following formula, I am not clear how to prove the second equation. I expand the left side of the second equation with the ...
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spectum of self adjoint operators

Let $H$ be an Hilbert space and $S = \displaystyle{ \sum_{i=1}^nS_i}$ where $S_i$ (i=1...n) is self adjoint with compact resolvent . is it true that the spectrum of $S$ is the sum of the spectra of ...
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relations between two linear operators

Let $\alpha,\beta$ be linear operators on a finite dimensional vector space $V$ over field $F$. Let $\gamma=\alpha\circ\beta$ and $\delta=\beta\circ\alpha$. Prove that: (1). $m_\delta(x)$ divides ...
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Limit of the spectrum in Banach algebra

Let $A$ an unital complex Banach algebra, $a_{n} $ is a sequence such that $\lim_{n\to \infty}a_{n}=a$. What is the relation between $\lim_{n\to \infty}\sigma(a_{n})$ and $\sigma(a)$. I think that it ...
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Multiplication operator on Hilbert space

i looked to the question Spectrum and point spectrum of this operator. I will go further with asking. We know that $T$ is well-defined iff $(\lambda_n)\in\ell^{\infty}$. But if ...
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Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal.

The question is: Prove that if $T$ a normal linear transformation and invertible, then $T^{-1}$ is normal. Then I have to find the spectral decomposition of $T^{-1}$. At first I tried to prove it by ...
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Singular value decomposition of an arbitrary anti-symmetric ($A=-A^{T}$) complex matrix

I am a physicist and very much used to the fact that any self-adjoint matrix ($H^{\dagger} =H$) in a finite-dimensional complex linear space can be uniquely specified by (a) the set of its (real) ...
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$T\in B(H)$ normal and left invertible implies $T$ invertible?

My question is what's written in the title, that is, if $T$ is a normal operator on a Hilbert space $H$, and $T$ is left invertible, is it necessarily true that $T$ is invertible? Actually, the more ...
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Why is this spectrum the closure?

I've started to learn about spectral theory and I'm looking at some examples. The spectrum is defined as $$ \sigma(a) = \{\lambda \in \mathbb C | a-1\lambda \text{ is not invertible} \}$$ If $A= ...
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spectrum of operators

Let $X$ be a Banach space and $T: X \rightarrow X$ is a bounded operator. The spectrum of $T$ is defined by $$\sigma(T) = \{\lambda \in \mathbb{C}; \lambda I - T ~~ \mbox{is not invertible}\}$$ It is ...
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Sources on Inverse Spectral Theory

I follow the book "Inverse Spectral Theory" by J. Pöschel and E. Trubowitz in which functional analysis is very much involved. I am curious about another approach using more real analysis and theory ...
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Approximate point spectrum and left topological zero divisors

Recall that a left topological zero divisor in a Banach algebra $A$ is an element $a\in A$ such that there exists a sequence of unit vectors $(a_{n})$ in $A$ with $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
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Spectrum of shift-operator

Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ...
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Relation of Hodge Theorem to Eigenfunction Basis of Laplacian

The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of ...
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Reason for Continuous Spectrum of Laplacian

For the circle $S^1$, it is well-known that the Laplace-Beltrami operator $\Delta=\text{ div grad}$ has a discrete spectrum consisting of the eigenvalues $n^2,n\in \mathbb{Z}$, as can be seen from the ...
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Spectral radius as the inf of norms of conjugates

I need help with the following problem: Let $A$ be a unital $C^{*}$-algebra. (a) If $r(a)<1$ and $b=(\sum_{n=0}^{\infty}a^{*n}a^{n})^{1/2}$, show that $b\geq 1$ and $||bab^{-1}||<1$. (b) ...
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Resolvent properties

Suppose that $A$ is a $n \times n$ matrix with $n$ different eigenvalues $\lambda_k.$ Corresponding eigenvectors are denoted as $x_k$, $x_k^Tx_k =1.$ Now $A=X\Lambda X^{-1}$. Denote $Q=X^{-1}$. ...
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Spectrum of a product

Let $A$ be a unital $C^{*}$-algebra. I am trying to show that if $a,b\in A$ are positive elements, then the spectrum of $ab$ is contained in the positive real numbers. I know that in the commutative ...
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Inverse of bounded self adjoint operator on HS is self adjoint?

Let $A=A^{*}$ be a bounded self adjoint operator on a Hilbert space $\mathcal{H}$ with Range Ran$(A) = D$ dense in $\mathcal{H}$. $A$ is injective, since Ran$(B) \perp ker(B^{*}) = ker(B)$. So ...
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Why these are equivalent?

Situation: operator theory, spectrum of a operator. We consider this as definition: $\lambda$ is a eigenvalue if $\lambda x=Tx$ for some $x\ne 0$ but I see someone saying this: $\lambda ...
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Estimate loss of information due to a low rank approximization by SVD

I have a matrix $X$ and I compute its Singular Value Decomposition: $$X = U \Sigma V^T$$ then, I take the lower rank approximization: $$X_k = U_k \Sigma_k V^T_k$$ where $k < rank(X)$, $U_k$ is made ...