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

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An inverse spectrum problem in linear algebra,

I am reading the book, Nonnegative Matrices, by Henryk Minc, and came across an exercise that I would like to solve: Let $$\bar \sigma = (\bar\lambda_1, ... , \bar \lambda_n)=(\lambda_1, ... \...
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Spectral radius as an upper bound to norm

The operator norm of a matrix is an upper bound for the spectral radius, and equality holds in particular when the matrix is Hermitian. One proof of this uses the spectral theorem. Is there a direct (...
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Proof compactness of adjoint operators

I am trying to understand a proof of the following statement: Given a complex B-space X and a compact operator $T:X\rightarrow X$, the adjoint operator $T^\ast:X^\ast \rightarrow X^\ast$ is compact as ...
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Fact regarding Kirchhoff's Theorem

Question regarding Kirchhoff's Theorem: If $ L(G)$ denotes the Laplacian of a graph $G$ then Kirchhoff's Theorem states that number of spanning trees in $G$ is equal to $(-1)^{i+j} \det L(i|j)$ ...
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Matrix -tree theorem-How to understand the theorem

I am having trouble understanding Kirchhoff's Theorem. The statement I want to prove is that if $\lambda_1,\lambda_2,...,\lambda _{n-1}$ are non-zero eigen values of $L(G)$ then Number of ...
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Inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$

I want to calculate the inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$ where, $w \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector and $ \alpha $ is some ...
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Compact operators form the only closed proper ideal of bounded linear operators

I am trying to understand the following proof in Trace Ideals and Their Applications by Barry Simon (Proposition 2.1): Let $\mathcal{J}$ be a two-sided ideal in $\mathcal{L}(\mathcal{H})$ containing ...
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Can $e^{ax}$ be said to be the eigenfunction of the operator $\frac{d^{(n)}}{dx}$?

I'm gradually getting familiar with operators (as they are used in QM) and the terminology surrounding them, and I was wondering whether all the (to me) well-known operators have straight-forward, ...
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What are eigenvalues/eigenfunctions of a “pointwise product” operator

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where "$\cdot$...
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Image density in spectral theory

The operator $T$ is $\dfrac{d}{dt}$ and $$\left\{\begin{array}{lc}x'(t)-\lambda x(t)=-y(t)\\ x(0)=0\end{array}\right.$$ and the domain of $T$ is $D(T)=\{x\in L^2(0,\infty):\; x\; \text{absolutely ...
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Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
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Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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23 views

What do Fourier Series for Other Symmetric Operators Look Like?

I understand that Fourier analysis works (up to constant multiples) by considering the inner-product space $E$ of smooth functions $[-\pi,\pi] \to \mathbb C$ with inner product. . . $\displaystyle (...
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Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
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Spectral theorem for unitary operators T o F

Si $T$ es unitaria y $B$ es una base de $V$ formada por vectores propios de $T$ entonces $B$ es un conjunto ortogonal. If $T$ is unitary and $B$ is a basis for $V$ consisting of eigenvectors of $T$ ...
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Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
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Uniqueness of spectral decomposition

Suppose $T: V\rightarrow V$ is diagonalizable on an arbitrary vector space (not necessarily an inner product space), so $T = \sum_{i=1}^r\lambda_i P_{\lambda_i}$ where $sp(T)=\{\lambda_1,\dots,\...
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Self-adjoint operators, projections, and resolutions of the identity.

In my Functional Analysis course, we're discussing the Spectral Theorem and the like. One question from a previous exam states the following: Let $H$ be Hilbert over $\mathbb C$, let $T \in B(H)$ be ...
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Example of operator with spectrum equal to $\mathbb{C}$?

In my Functional Analysis course, we proved that for a (possibly unbounded) operator $T$ that is densely defined, closed, and symmetric, exactly one of the following four occurs: $\sigma(T) = \...
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Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
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Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator (...
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Spectrum of operator $T((x_n)_{n\in\mathbb{Z}})=\left(\frac{1}{n^2+1}(x_n-x_{-n})\right)_{n\in\mathbb{Z}}$

The eigenvalues should satisfy: $$T(x_n)=\lambda x_n$$ $$\frac{1}{n^2+1}(x_n-x_{-n})=\lambda x_n$$ $$\left[(n^2+1)\lambda+1\right]x_n=x_{-n}$$ I suppose that this should mean that $$\forall\lambda\in\...
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Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually I'm not exactly looking into bipartite but the ...
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Can one hear the *material* of a drumhead?

"Can one hear the shape of a drum?" is a well known problem, originating from Kac, 1966, that questions whether an (idealized) drum head is completely specified by its spectrum. That is: is the ...
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Spectrum of operator in $l_2$

Sorry guys, I have a problem with this exercise. Let T an operator in $l_2$ Hilbert space: $$ (\textrm{T}x)_1 = x_2 , $$ $$ (\textrm{T}x)_2 = 0 , $$ $$ (\textrm{T}x)_n = x_{n-1} - x_n $$...
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How to prove that $f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda$ where $f_{\lambda} $ is the eigenfunctions of $\Delta$

On Euclidean space $\mathbb R^n$, how to prove that $$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$ where $\Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) $, whith $\Delta$ is the ...
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References on the spectral theory of the Floquet problem

The Floquet problem is about the linear ordinary differential equation $$ \dot{\psi} = A(t) \psi. $$ Here $A(t)= A(t+T)$ is a periodic $n\times n $ matrix. Suppose $A(t) =-i H(t)$ with $H(t)$ being ...
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Proof: the resolvent operator is holomorphic.

I tried to prove that the resolvent operator $$\rho(A) \to \mathbb C,\space \lambda \mapsto R_{\lambda}(A):=(\lambda id_X -A)^{-1} $$ is holomorphic, where here $A$ is a bounded linear operator from ...
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Nonempty intersection between approximate point spectrum and residual spectrum

On the Wikipedia page on "Spectrum (functional analysis)", it is mentioned that the approximate point spectrum and residual spectrum are not necessarily disjoint. Is there a straightforward example to ...
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Different versions of Mercer's theorem

I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem: About the positive-definiteness conditions. In the Wikipedia pages on RKHS ...
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Spectrum $\sigma(T)$ of $T:l^1 \to l^1$ given by $T((a_j))=\left( \sum_{j=2}^{\infty} a_j \right) e_1 + \sum_{j=2}^{\infty} a_{j-1} e_j$

I'm considering the bounded linear operator $T$ on $l^1$ (the space of all absolutely convergent complex sequences) given by (with $e_k=(\delta_{kj})_{j=1,2,...}$) $$T((a_j))=\left( \sum_{j=2}^{\...
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Classify sub $C^*-$ algebras of $\mathbb{C}^{2 \times 2}$

Apparently if $A$ is a sub $C^*-$ algebra of the complex $n \times n$ matrices then we can characterize these subalgebras as block matrices.Now, for the case $n=2$ I was wondering if there is an easy ...
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Uniform convergence in Mercer Theorem for bounded kernels

Let $\mu$ be a finite, strictly positive measure on $\mathbb{R}$, and let $k$ be a measurable positive-definite kernel. Assume $k$ is bounded, and let $T:L^2(\mu)\rightarrow L^2(\mu)$ be defined by $$ ...
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Orthogonal projection onto the eigenspace of compact, self-adjoint operators.

Let $T$ be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that $f\in H$, $||f|| =1$ and $||(T-3)f||\leq 1/2$. Let P be the orthogonal projection onto the direct sum of all ...
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Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it.
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The topology of the spectrum of a linear operator

In general a spectrum of a linear operator has a decomposition into three parts: point spectrum, continuous spectrum and residual spectrum. What I'm interested in is the topology of these parts of ...
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Clarification on point spectrum of an operator

From my understanding, if $\lambda$ is in the point spectrum, then $\lambda$ is a complex number such that it satisfies the equation $(T - \lambda I) x = 0$. My confusion arises from problems like ...
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Isolated Eigenvalues on the Extensions.

I asked this question on Mathoverflow http://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer ...
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$\lambda$ is an eigenvalue iff spectral measure of $\lambda$ is nonzero

Let $M$ be a normal operator on a Hilbert space and let $E$ be the spectral measure of $\sigma(M)$ (the spectrum of $M$). Show that $\lambda$ is an eigenvalue to $M$ $\iff E(\{\lambda\})\not = 0$. ...
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Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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Spectrum of right shift operator in weighted $l2$ sequence space

Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
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The k-th derivative of the resolvent set

I want to prove $$\frac{d^{k}}{dz^{k}}(zI-A)^{-1}=(-1)^{k}k!(zI-A)^{-k-1}$$ I have the resolvent equation $(zI-A)^{-1}-(\lambda I-A)^{1}=(\lambda-z)(zI-A)^{-1}(\lambda I-A)^{-1}$, i.e. $$\begin{...
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Spectrum of the laplacian on a Banach space

Is the spectrum of the laplacian on $L^1(0,1)$ with Neumann boundary conditions known?
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Show that an operator is negative

I would show that, the operator $$A = \left(x_{4} \frac{\partial}{\partial x_{1} } -x_{1} \frac{\partial}{\partial x_{2} } \right) \frac{\partial}{\partial x_{3} } $$ is a negative operator on $\...
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Is $f$ unitary if $f\in L(V,V)$ such that $a<\frac{\|f^n(v)\|}{\|v\|}<b$, with $0<a<1<b$, for all $n\in \mathbb{N}$ and nonzero $v$

Let $V$ be a complex inner product space. Is $f$ unitary if $f\in L(V,V)$ such that $a<\frac{\|f^n(v)\|}{\|v\|}<b$, with $0<a<1<b$, for all $n\in \mathbb{N}$ and nonzero $v$? If not, ...
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If $0\leq a \leq b$ and $a$ is invertible, then $b$ is invertible

Let $\mathscr A$ be a unital C*-algebra and let $a,b\in \mathscr A$ such that $0\leq a \leq b$ and $a$ is invertible. How to show that $b$ is invertible? ($0\leq a \leq b$ means that $a,b$ is ...
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Spectrum radius upper bound of Hadamard product

I am doing some research that relies on Hadamard product of two matrices bound, the most famous one that I encounter is : $\rho(A\circ B)=\rho(A)\rho(B)$ this seems to be trivial when I test it with ...
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Positive logarithm in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible. I just see that the usual $\log$ function is continuous on ...
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What does the spectrum of the adjacency matrix of a graph tell you? [duplicate]

I am trying to search for an answer to the following question and I cannot find a straightforward answer. What does the spectrum of the adjacency matrix (set of eigenvalues and their multiplicities) ...
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T be compact operator defined on $L^2(\Omega)$ Show that the null space of T-I satisfies

Can anyone help me out on this one Let $$ \Omega\subset\text{R^d be a domain K}\,\in\,L^2(\Omega X \Omega)$$ and T be compact operator defined on $L^2(\Omega)$ by Tf(x)=$\int_\Omega K( x,y)\text{f(...