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

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Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
2
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1answer
163 views

Cayley Transform: well defined?

Why is the Cayley backtransformation well-defined: $$A_U:=\imath(1+U)(1-U)^{-1}$$ In general $1-U$ is not invertible for example for $U=1$.
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Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
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93 views

Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...
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163 views

What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
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1answer
39 views

Inequality norms

Let $A$ be a bounded linear operator on a Banach space $X$. Can we show that for an arbitrary $n \in \mathbb{N}$ and $x \in X$ such that $\|x\|_X \geq 1$ we have that $$\|A^n x \| \leq \|Ax\|^n.$$ ...
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746 views

Resolvent: Definition

Given a Banach space. Consider linear operators: $$T:\mathcal{D}(T)\to E:\quad T(\kappa x+\lambda y)=\kappa T(x)+\lambda T(y)$$ (No other assumptions on the operator!) Denote for shorthand: ...
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42 views

Spectral theory - how to prove this lemma?

in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3): If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda ...
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2answers
80 views

Operators $A$ such that $e^A$ is norm preserving

Let $X$ be a Banach space. $A$ a bounded operator. We can define the exponential of $A$ by $$e^{A}=\sum_{n=0}^{+\infty}\frac{A^n}{n!},$$ which is also a bounded operator. Is there any sufficient ...
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3answers
105 views

Finding spectrum of the operator A

$A:\mathcal{l}_2\rightarrow \mathcal{l}_2:(x_n)_{n=1}^\infty \rightarrow (x_{n+1})_{n=1}^\infty$ (left shift) Find the spectrum and all its parts for the operator A. What should I do?
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56 views

How to show whether this operator is normal? self-adjoint? unitary?

Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$ How to show whether this operator is normal? self-adjoint? unitary? ...
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1answer
29 views

How to show that the spectrum is equal to the range of $y$

How to show that the spectrum of $T_y$ is equal to the range of $y$ Given $y\in C[0,1]$ and $T_y: C[0,1] \rightarrow C[0,1]: x\mapsto x\cdot y$ Any help is appreciated, thanks.
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28 views

When can we get discrete spectrum?

Suppose that $T$ is a densely defined closed operator on a separable Hilbert space $H$. Form $N = T^*T$. Assume further that $T$ has a finite dimensional kernel and satisfies the commutation relation ...
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1answer
37 views

When $A_y$ is invertible?

Given $y\in C[0,1]$ Let $A_y:C[0,1]\rightarrow C[0,1]: x\mapsto xy$ When $A_y$ is invertible? Could you please help.
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1answer
44 views

How to find the point spectrum $\sigma_p(A)$ of $A$

How to find the point spectrum $\sigma_p(A)$ of $A$ Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$ ...
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116 views

eigenvalues to Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
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1answer
128 views

Help please eigenvalue of Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2\left(\Omega \right)$. Let $\left(\lambda_n\right)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and ...
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109 views

Doubt about integration over a curve

I'm reading a paper and I have a doubt: Suppose we have a compact Riemannian manifold and an eigenfunction of the Laplace operator $\Delta u=-\lambda u$. Suppose in addition that we have the following ...
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1answer
70 views

Positive Operator: Norm Estimate

In class we encountered the statement: $$H\geq C1\quad(C>0)\implies\|\mathrm{e}^{-\beta H}\|<1\quad(\beta>0)$$ How does one prove this? Moreover, what about the weakened version: $$H\geq ...
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58 views

Question with tried Eigenvalues of Laplacian operator and Sobolev spaces III.

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
2
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1answer
57 views

Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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55 views

How to find the spectrum $\sigma_p(P)$

How to find the spectrum $\sigma_p(P)$: Let $P:H\rightarrow H$ be an orthoprojection, $P\neq 0, P\neq I$. could you please help
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2answers
74 views

How to show: $A_y$ has no eigenvectors if $y$ is not constant on any subinterval of $[0,1]$

Let $y\in C[0,1]$ and $A_y : C[0,1]\rightarrow C[0,1]: x\mapsto xy$ How to show: $A_y$ has no eigenvectors if $y$ is not constant on any subinterval of $[0,1]$. Could you please help.
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1answer
35 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
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253 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
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1answer
51 views

Question on the Spectral Theorem.

If $A:D(A)\subseteq H\to H$ is a densely defined self-adjoint operator, we can make sense of the unitary group $e^{itA}$ with the Borel functional calculus. I'm really struggling to understand why ...
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1answer
50 views

Prove that, if $R_\lambda$ is resolvent fuction and $\lambda>0$, then $\|R_\lambda\|\leq \frac{1}{\lambda}$

Let $T$ be a linear operator defined on a limited complex Banach space $X$. Resolvent set of T is defined as the set $ \rho (T) $ of complex numbers $\lambda$ such that the operator $ \lambda I - T $ ...
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103 views

Left and right eigenvectors perpendicular to each other

I just read in a textbook on numerical methods that you can always have that the right eigenvectors of a matrix can be taken as orthonormal to the left eigenvectors for a diagonalisable matrix. This ...
3
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1answer
86 views

Want to show that an operator is not surjective

So here is my problem, Let $$M_1:L^1\rightarrow L^1$$ $$f(x)\mapsto \arctan(x)f(x)$$ In order to compute the spectrum of $M_1$ I am investigating for which $\lambda\in\mathbb C$ the following map is ...
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155 views

Comparison of versions of the spectral theorem

Definition: (Resolution of Identity) A projection valued measure that is finitely additive and factors over intersections. Theorem 1: Let $A$ be a normal operator in $H$ a Hilbert Space where ...
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114 views

Generalized eigenvalue problem; why do real eigenvalues exist?

Under this text, you can see two pairs of matrices $(A,C)$. I am currently solving the generalized eigenvalue $(A-\lambda C)v=0$ for several pairs of $(A,C)$ and found out that they also have real ...
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Polynomial Calculus: well defined?

Disclaimer This thread has been refreshed! Problem Given a C*-algebra $\mathcal{A}$. Consider an element $A\in\mathcal{A}$. Introduce an abstract polynomial calculus: $$A=A^*:\quad ...
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75 views

Polynomial Ring: Root vs. Remainder

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
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1answer
71 views

Establishing an inequality between the $2^{nd}$ largest eigenvalue of $A$ and a related matrix.

Let $A$ be an irreducible, aperiodic matrix with non-negative entries, with $1 \in \ker(A - I)$, $w \in \ker(A^\top - I)$, $w_i > 0$ $\forall i$. Define $W = \text{diag}(w)$. I am studying the ...
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1answer
60 views

Is it possible to consider an approximation to a (non-self adjoint) operator with a self adjoint one?

In operator theory it's wonderful if we have a self-adjoint operator (non necessarily bounded) due to all the work that has been done using their symmetry,... etc. I.e there are many powerful tools. ...
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2k views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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830 views

Condition for degenerate eigenvalues for a matrix

Given a diagonalizable matrix $M$ (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ...
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113 views

Examples of spectrum of compact operators on the sequence space $l_2$

Suppose $T$ is a compact operator on the sequence space $l_2$, and let $\sigma(T)$ be its spectrum. Is it possible to find a $T \ne 0$ such that $\sigma(T) = \{0\}$? Also, is it possible to find $T$ ...
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56 views

Resolvent of the differentiation operator on the torus

Suppose we have the space $L^2(\mathbb T)$, that is, the space of periodic functions that are in $L^2$. Let $L^0$ be the operator of differentiation, i.e $L^0 f = f'$ where the domain of $L^0$ is the ...
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1answer
141 views

Multiplication operator has no no eigenvalues

Statement: Let $M_x$ denote the multiplicative operator acting on $L^2([0,1], \, dx)$ by $M_x(f) = xf$. Show that $M_x$ has no eigenvalues Attempt: Let $M_x(f) = xf$ then we should have $M_x(f) = ...
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Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
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804 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
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stability of essential spectra

Let $X$ be a Banach space. $A$ and $B$ are linear closed and densely defined operators and $\lambda\in\rho(A)\cap\rho(B)$ such that $(\lambda - A)^{-1}-(\lambda - B)^{-1}$ is a Frehholm perturbation ...
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A question about matrix spectrum property

Suppose $x\in\mathbb{R}^n$, $A\in\mathbb{R}^{n\times n}$. Does anyone know the answer to the following problems. (1) $\min\limits_{x\neq0} f(x)=\frac{x^\mathrm{T}A^\mathrm{T}Ax}{x^\mathrm{T}Ax}$, ...
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1answer
102 views

How do I show a left inverse of a bounded linear operator on Banach space?

If $A$ is a bounded linear operator on a Banach space X, with a left inverse $A_l^{-1}$, and P is a projection (also on X), how do I show that $A_l^{-1}P$ is also a left inverse of A (i.e. ...
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201 views

Power series convergence of random walk transition matrix

I would like to find out if $$ \sum_{t=0}^\infty P^t = \left( I- P \right)^{-1} ~,$$ where $P = D^{-1}W ~ $ is a random walk transition matrix. $W$ is a matrix describing weights in a graph and ...
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1answer
64 views

Eigenvalues of adjoint for residual spectrum.

Statement: Let $T$ be a bounded operator in a Hilbert space $\mathscr{H}$ Show that if $T-\lambda I$ is not dense in $\mathscr(H)$, then $\overline{\lambda}$ is an eigenvalue of $T^*$. Attempted ...
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2answers
68 views

Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho ...
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Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
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What are these spectra (part 2)

I needed some insight into spectra in Banach algebras. To this end I tried to calculate a few examples. I will post them not all in one post. Could somebody please tell me if they are correct? Here ...