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

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Eigenvalues of polynomials of a matrix and its inverse up to summation by identity

There is a paper that I am reading and the following has been considered without proof: (Suppose $\lambda(.)$ defines the spectrum of a matrix and one can define a random variable on this spectrum say ...
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59 views

Spectrum of a bounded operator $T$ satisfying $T^n=I$

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, suppose $T\in \mathcal{B}(\mathcal{H})$ is a bounded operator and suppose that $n$ is the smallest natural number so that $T^n=I$. Let ...
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38 views

A little question about using spectral mapping theorem for polynomials

A question from Kreyszig: Let $X$ be complex Banach space with $T\in B(X,X)$ and a $p$ a polynomial. Show that the equation $p(T)x=y$ has a unique solution $x$ for every $y \in X$ if and only if ...
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63 views

Spectral radius of an operator equals its norm

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. We know that the spectrum of $A$ is always included in the ball $B(0,|A|)$ and the spectral radious $r(A)$ is the smalest radius such that ...
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66 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...
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Spectrum of the sum on a tensor product?

I have the following problem. Consider the operator $R= H\otimes 1 + 1 \otimes K$ on the tensor product $\mathcal H \otimes \mathcal K$ where $H$ and $K$ are self-adjoint. I know that $R$ has a ...
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63 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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84 views

A decomposition of Hilbert space via self-adjoint operator

Let $H$ be a complex Hilbert space and $A:H\to H$ self-adjoint. Show that one can decompose $H$ into two $A$-invariant closed subspaces as $H=H_{p} \bigoplus H_{c}$ such that the spectrum of ...
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44 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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16 views

Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
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78 views

Selfadjoint Operator: Basic Criterion

For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way?
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What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
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2answers
129 views

Book: Functional Calculus

Is there a good book that investigates in detail the various kinds of functional calculus? I'm having now some knowledge about unbounded operators and integration but I would like to understand ...
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100 views

Application of eigenvalueproblems for the wave equation

I'm currently searching for a nice little application of an eigenvalueproblem and found the following for acoustics - but one part doesn't make sense for me. Consider the wave equation to find some ...
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112 views

“right shift” il $L^1$

Let $X=L^1(\mathbb{R})$ be the space of Lebesgue integrable functions $f:\mathbb{R}\rightarrow \mathbb{C}$ with the usual norm. Let $T\in B(X)$ be defined by $$(Tf)(t)= f(t+1)$$ I need to find the ...
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33 views

Why is this subspectrum closed

Let $u: X \to X $ be a compact operator on a Banach space $X$ and let $\lambda \in \mathbb C$ be non zero. We know that $u-\lambda$ is Fredholm and that $X=\mathrm{ker}(u-\lambda)^n \oplus ...
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109 views

Symmetric Operator vs. Real Spectrum

For symmetric operators one has a characterization: $$A\text{ symmetric}:\quad A=A^*\iff\sigma(A)\subseteq\mathbb{R}$$ (I want to investigate to what extend symmetry is a necessary assumption.) ...
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122 views

Normal Operators: Spectrum vs. Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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2answers
96 views

Adjoint of sum of two operators

Let $A$ be self-adjoint and $B$ symmetric (which means densely defined for me as well) with $A$-bound less than $1$. Does this imply that $(A+iB)^*=A-iB$ ?
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74 views

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 ...
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87 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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80 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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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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37 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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283 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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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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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
95 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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45 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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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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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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36 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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35 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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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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104 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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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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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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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 ...
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53 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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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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eignvalues of laplacian operator and Sobolev spaces -II

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, Let $F=(F_t) \in C^0(I,L^2(\Omega))$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od the ...
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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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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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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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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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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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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 ...
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70 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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117 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 ...