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

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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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73 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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120 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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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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71 views

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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67 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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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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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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861 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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385 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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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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47 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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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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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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65 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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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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48 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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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 ...
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the spectrum of a matrix

If $A$ is an $n \times n$ nilpotent matrix show that $I-A$ is invertible then find the spectrum of I-A ? for part one i've shown that $I-A$ is invertible by finding its inverse using that $A$ is a ...
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Time-independent magnetic Schrodinger equation

Let $B$ be a magnetic field, let $\mathbf{a}$ be a vector and let $\Psi$ be the wave function. If $\mathcal{H}(B) = (-i\nabla + B\mathbf{a})^2$, where $\nabla$ is the gradient, then the ...
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interpretation of generalized eigenvalue/vectors in spectral graph theory

Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...
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Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
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Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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In which cases the spectrum of an operator contains only eigenvalues?

Let $X\neq \{0\}$ be a complex normed spaces (not necessarily finite-dimensional) and $T:D(T)\subset X\to X$ a linear operator (not necessarily bounded). I would like to know under what conditions can ...
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Application of the spectrum of an operator

http://en.wikipedia.org/wiki/Spectrum_of_an_operator What is the application of the spectrum of an operator
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Spectral Mapping Theorem

Spectral mapping theorem is as follows: https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf Is Spectral mapping theorem true for point spectrum ?
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Interpretation of Power Spectral Density (DTFT of Covariance function)

If we have a deterministic signal $x[n]$ and its transform $$ X(f) = \sum\limits_{n=-\infty}^{\infty}x[n]\exp\left(-2\pi fn\right)$$ I can think of this as containing knowledge of a discrete-time ...
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Spectrum of idempotent element

Let $A$ be some unitary algebra over $\mathbb{C}$. If $a^2=a$ and $0\ne a\ne 1$ then $\{0,1\}\subset \sigma_A(a)$ ($\sigma_A(a)$ is the spectrum if $a$). I believe that also $\sigma_A(a)\subset ...
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Spectral theory and sequences: is this fact a general truth or does it depend on the operator?

Let $\lambda\in\mathbb{R}\setminus\{0\}$, $\textbf{i}$ the imaginary unit, $H$ a Hilbert space, $T:D(T)\subset H\to H$ a invertible densely defined linear operator such that $T^{-1}$ is bounded, ...
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Rotating the spectrum of a bounded operator

If $T$ is a bounded operator on a Banach space $X$, and $\sigma(T)$ is its spectrum, what would be an operator whose spectrum is $\sigma(T)$ rotated by $\theta$? For example, $-T$ has as spectrum ...
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Any way to simplify this expression?

So I have a vector of asset allocation weights given by $x \in R^4$ and a covariance matrix of the asset returns $\Sigma \in R^{4,4}$. I know by the spectral theorem, $\Sigma = V DV^{-1}$ and the ...
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Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
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Rudin's proof of invariant subspace existence

I have questions about Rudin's proof of invariant subspace existence. On page 327, point 12.27, How does he get that $Tx=TE(\omega)x$, and How does he know $E(\omega)$ is not the zero map? ...
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Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
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What is a good book that focuses on the applications of complex analysis and spectral theory?

My research involves a great deal of complex analysis and spectral theory, and I always feel a bit flustered when non mathematicians ask me what I study. It's hard to explain the math in layman's ...
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The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
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Adjacency matrix of directed graph

I am given adjacency matrix $A$ of directed graph. $A(x,y)$ counts the number of edges from $x$ to $y$. I want to show that if $A$ has constant outdegree $d$: (i) For any eigenvalue $\lambda$, we ...
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How to use Parseval' identity( Plancherel)? [duplicate]

(May be this is very basic question for MO) Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put $$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$ Fix ...
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Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
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How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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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) ...