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

learn more… | top users | synonyms (1)

1
vote
1answer
76 views

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 ...
1
vote
0answers
218 views

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) ...
2
votes
1answer
37 views

$T\in B(H)$ 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 ...
0
votes
1answer
37 views

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= ...
1
vote
1answer
63 views

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 ...
2
votes
0answers
48 views

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 ...
2
votes
1answer
44 views

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 $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
3
votes
1answer
218 views

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 ...
1
vote
1answer
116 views

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 ...
6
votes
4answers
196 views

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 ...
0
votes
0answers
30 views

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) ...
0
votes
1answer
52 views

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 $Q=X^{-1}$. ...
0
votes
1answer
77 views

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 ...
1
vote
0answers
115 views

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 ...
1
vote
2answers
45 views

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 ...
0
votes
0answers
84 views

Find the spectrum of this operator

$$O:l_p\to l_p\quad p\in[1,\infty]$$ $$Ox=(0,x_1,x_2,x_3,...)\quad \forall x=(x_1,x_2,x_3,...)\in l_p$$ if $\lambda\ne0$ then we have $$\lambda x-Ox=(\lambda x_1,\lambda x_2-x_1,\lambda ...
3
votes
1answer
39 views

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 ...
2
votes
1answer
62 views

Closure of numerical range contains spectrum

Let $A: D(A) \subset \mathcal{H} \to \mathcal{H}$ be a densely defined operator on a Hilbert space $\mathcal{H}$ with adjoint operator $A^{*}$. Given that $D(A) = D(A^{*})$ I'm trying to show that the ...
0
votes
3answers
67 views

Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
2
votes
2answers
127 views

Problem with spectral theorem and spectral measure.

There is a passage in a book that is not very clear to me: A is a C*Algebra and $a$ is selfadjoint. Then "Indeed identifying A with an algebra of operators on a Hilbert space $\mathcal{H}$, by the ...
0
votes
0answers
33 views

A domain in which the dirichlet laplacian has eigen values of all orders

I am trying to come up with an example to the following. Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., ...
1
vote
0answers
39 views

Fourier Transform of Fractional Laplacian

I'm trying to solve a PDE with a spectral method. The PDE has a fractional Laplacian... $\Delta^s$. In regards to a numerical implementation, will the "s" term simply become the exponent of the ...
0
votes
1answer
28 views

Strict inequality in spectral radius estimate

I am interested to find an example which shows strict inequality in the spectral radius estimate that is $r(x) \leq \|x \|$ . I would like to see when $r(x) < \|x\|$.Normal operator is the case ...
4
votes
1answer
80 views

Eigenvalues of the operator $(Tu)(x)=\int_0^x (\int_t^1 u(s)ds)dt.$

Consider the linear operator $T$ in $L^2(0,1)$ defined by: $$(Tu)(x)=\int_0^x \left(\int_t^1 u(s)ds\right)dt.$$ I have managed to prove that it's continuous,self adjoint,compact but now I have to ...
1
vote
0answers
39 views

Properties of the operator $ (Tu)(n)=u(n+1)-u(n-1)$

Let $T:l^2(Z,R)\to l^2(Z,R)$ the linear bounded operator defined by: $$ (Tu)(n)=u(n+1)-u(n-1)$$ a)Prove that the image of T is dense. b)Prove that $\forall \lambda\neq 0\quad t-\lambda I$ is ...
2
votes
1answer
34 views

Does non-Hermitian implies at least one complex eigenvalue?

Ok so I'm studying linear algebra and we went trough the Spectral Theorem, including and proving the fact that for every Herimitian matrix, its eigenvalues have $0$ imaginary parts. I was wondering is ...
3
votes
1answer
70 views

Connections between Lebesgue-Radon-Nikodym decomposition and spectral decomposition

Perhaps this is a silly question to ask but it's been on my mind for a bit. When I took my first course in functional analysis a year ago, we covered spectral theory. Particularly, we covered spectral ...
1
vote
1answer
320 views

Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
2
votes
1answer
102 views

Eigenvalues and adjoint of operator $T(x_k)_{k=1}^{\infty} = (x_{2k})_{k=1}^{\infty}$

Let $T$: $l^2 \rightarrow l^2$ denote the operator \begin{align} T(x_1,x_2,\dots, x_n,\dots) = (x_2,x_4,\dots,x_{2n},\dots). \end{align} There are several questions regarding this operator that I need ...
1
vote
1answer
85 views

Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
0
votes
1answer
69 views

Square root in Banach algebra

Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin ...
0
votes
1answer
535 views

The eigenvectors of a matrix and its transpose that correspond to the same eigenvalue are not orthogonal

Spent hours trying to prove this after encountering it in Lax's discussion of the spectral theorem, but no luck. Here's the problem (it is Theorem 18 in Lax 2ed, Chapter 6): A mapping $A$ has ...
3
votes
1answer
65 views

Spectrum of Symmetrizable Matrix

A matrix $ M $ is symmetrizable if $ M = D S $ with $ D $ a square diagonal matrix with positive entries, and $ S $ a symmetric matrix. What can be said about the spectrum of $ M $? It seems like I ...
0
votes
2answers
91 views

Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
0
votes
1answer
34 views

“Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ...
2
votes
1answer
60 views

Spectral theorem for $n$-tuples of selfadjoint operators

I need a 'good' reference to the following version of the Spectral Theorem: Given $n$ commuting selfadjoint operators on an infinite-dimensional Hilbert space, there exist a Borel measure $\mu$ on ...
2
votes
1answer
109 views

expectation of norm of orthogonal projector

The question has to do with calculating the expected squared norm of a random projection. We have a 2D subspace $T := span\{U1, U2\}$ where $U1$ is a random vector uniformly distributed over unit ...
3
votes
1answer
63 views

equality of two operators…

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two ...
0
votes
1answer
55 views

Proof of a Lemma guaranteeing the existence of the Borel-measurable functional calculus

In my lecture I had the following Lemma, which guarantees the existence of the Borel-measurable functional calculus: Le $(H,<,>)$ be a complex Hilbert space and let $q:H\rightarrow \mathbb{C}$ ...
1
vote
1answer
61 views

Using Weyl sequences to prove relation between quadratic form and spectral radius

I know that the formula $$\lVert A\lVert=\sup_{\lVert x\lVert=1} \langle x,Ax\rangle$$ holds true for self adjoint operators. While reading Teschl's book I saw a comment that on can prove this formula ...
0
votes
0answers
25 views

Time Series: Spectral Density

Prove that $$ \int_{-\pi}^\pi e^{i(k-h)λ} dλ = 0 $$ if k!=h solving the integral gives $$ \frac{2 sinh(\pi * i(k-h))}{i(k-h)} $$ I don't see how this will give 0 with k!=h (also if k=h, the ...
2
votes
1answer
125 views

Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...
3
votes
1answer
132 views

A simple question about *-homomorphism in C*-algebra

Let $A$ and $B$ be C*-algebra, $h\colon A\rightarrow B$ is *-homomorphism. If $a\in A_{\operatorname{sa}}$, then $\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. ...
1
vote
1answer
133 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
0
votes
0answers
37 views

Do eigenfunctions from symmetric ODEs really make up a base for $L^2$?

I'm confused about a theorem in some hand-out material on spectral theory for ODEs. The BVP is stated in the form $$ \begin{cases} L u = f\\ R u = 0 \end{cases} $$ over $I = [a,b]$. And ...
2
votes
0answers
67 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
0
votes
2answers
63 views

if T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎

I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎ Its proof is obvious from one hand.But i cant prove that ...
3
votes
1answer
109 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
4
votes
0answers
86 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
2
votes
0answers
64 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...