Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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36
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2k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
11
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510 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
7
votes
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410 views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
6
votes
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94 views

When does analytic in the operator norm imply analytic in the trace class norm?

Consider $U$ a nice compact region in $\mathbb{C}$ with boundary $\Gamma$. Let $S_1$ b the ideal of trace class operators on a separable complex Hilbert space $H$. We will let $\|\cdot \|$ be the ...
6
votes
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241 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
6
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178 views

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 ...
6
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255 views

Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...
6
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386 views

Hilbert transform and Hilbert matrix

The Hilbert matrix is \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt] ...
6
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0answers
565 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
6
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434 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
6
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168 views

Invertibility of Toeplitz operator in $\ell_1$

Suppose we have a Toeplitz operator $$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & ...
6
votes
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231 views

Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting

The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below. Suppose $M$ is a smooth Riemannian manifold ...
5
votes
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61 views

Infer $Tf=\sum_{n=1}^\infty (f,f_n) f_n$ from frame condition.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in H$ ...
5
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34 views

Spectrum of periodic schrödinger operators

In many articles it's stated, as if it's common knowledge, that any Schrödinger operator with periodic potenial has purely absolutely continuous spectrum. I've tried to actually find a theorem ...
5
votes
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54 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
5
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64 views

Check proof about range of bounded linear operator.

I have to prove that the range $\mathcal{R}(T)$ of bounded linear operator $T:X\rightarrow Y$; $X,Y$ normed spaces need not be closed in $Y$. As a hint I'm given that I could consider ...
5
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28 views

Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...
5
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59 views

Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space ...
5
votes
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55 views

How can I show that given a norm one linear functional on $c_0$ that there is a unique extension to a norm one functional on $\ell_\infty$?

We are given that our Banach space is $c_0 \subset \ell_\infty(\mathbb{N})$ and there is a functional $y^* \in c_0^*$ such that $||y^*|| = 1$. We are guaranteed that this extends, via Hahn-Banach to a ...
5
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206 views

Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
5
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173 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
5
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442 views

Sum of operator and adjoint is self-adjoint

In abstract Hodge theory there is the following lemma: Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
5
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422 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
5
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161 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
4
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93 views

Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
4
votes
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69 views

Compute the spectrum of the integral operator $K:L^2([0,1]) \to L^2([0,1])$ defined as $(Kx)(t) = \int_0^t x(s) ds$

Let $K:L^2([0,1])\rightarrow L^2([0,1])$ be the linear operator defined by $$(Kx)(t)=\int_0^tx(s)ds, \quad x \in L^2([0,1]).$$ Now I have to compute the spectrum, but I don't have any idea how to do ...
4
votes
0answers
54 views

is the pullback operator associated to a flow bounded in L^2?

Let $M$ be a smooth compact manifold with a finite Borel measure $m$. Let $\{f_t\}_{t\in\mathbb R}$ be a $C^1$ flow on $M$. That is, a $C^1$ function $$ \mathbb R\times M\ni(t,x)\mapsto f_t(x)\in M $$ ...
4
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129 views

On the exponential form of a unitary matrix

A unitary matrix $U \in \large C_{n,n}$ can always be written in the exponential form: $U = e ^{iA}$ (1) where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary ...
4
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85 views

Trace class operators problem

Let $\mathcal{B}_1(\mathcal{H})$ be the set of trace class operators in a Hilbert space $\mathcal{H}$ and $\mathcal{H}^{(d)} = \bigoplus_{i=1}^d \mathcal{H}$ with $1 \leq d \leq \infty$. If $C \in ...
4
votes
0answers
112 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
4
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0answers
132 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
4
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0answers
111 views

How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators $Ran(A)$ is closed, and $\dim(\mathrm{Ker}(A))$ or ...
4
votes
0answers
115 views

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 ...
4
votes
0answers
52 views

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 ...
4
votes
0answers
190 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
4
votes
0answers
112 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 = ...
4
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0answers
110 views

Concerning unbounded linear operators on a Hilbert space

Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
4
votes
0answers
511 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
4
votes
0answers
231 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
4
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82 views

The control of norm in quotient algebra

Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and ...
4
votes
0answers
444 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
4
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185 views

surjectivity of operators on $\ell^\infty$

Can anyone give me an example of an operator $T:\ell^\infty\rightarrow\ell^\infty$ which has dense range but is not surjective?
4
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129 views

Does such an operator exist?

Suppose $T$ is a bounded operator on $H:=\mathcal{l}_2$ which is quasinilpotent and has the property that both $T$ and $T^{*}$ are not injective and have finite dimensional kernels. Is it possible ...
3
votes
0answers
41 views

Physical meaning of Rudin's equation in Hilbert space

Rudin's Functional Analysis, p. 334, Corollary of Theorem 13.10 says Corollary If $a\in H$ and $b\in H$, the system of equations $$-Tx+y=a$$ $$x+T^*y=b$$ has a unique solution with $x\in ...
3
votes
0answers
44 views

Evaluate the norm of a linear operator

Can someone help me on this question. I want to compute the norm of the following operator $$l:\mathbb R^N\longrightarrow \mathcal M_n(\mathbb R); (x_1,\cdots,x_N)\mapsto (x_i - x_j)_{1\leq i,j\leq ...
3
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55 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
3
votes
0answers
37 views

Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. ...
3
votes
0answers
35 views

Why is the Calkin algebra purely infinite?

I tried using the fact that in a simple unital $C^*$-algebra, $\mathcal{A}$, purely infinite is equivalent to the following: If $x\in\mathcal{A}$ is non-zero, then there exists $a,b\in\mathcal{A}$ ...
3
votes
0answers
42 views

Relating Fourier transform theory on two distinct subspaces

In Fourier transform theory (on $\mathbb{R}$), three vector spaces play a very important role: $L^1(\Bbb R)$, $L^2(\Bbb R)$ and the Schwartz space $\mathcal{S}(\Bbb R)$. Arguably the nicer spaces of ...
3
votes
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69 views

Perturbation of Laplacian

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...