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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Forward difference operator

What does $\Delta^{-1}$ mean? I have seen it in a question such as "justify that $\Delta^{-1}k^{(n)} = {k^{n+1}\over{n+1}}$". Thanks for your help.
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$\omega$ is cyclic for $M\subset B(H)$ if and only if $\omega$ is separating for $M'$

Let $H$ be a Hilbert space, $M\subset B(H)$ a von Neumann algebra and $\omega \in H$ a vector. Then $\omega$ is cyclic for $M$ if and only if $\omega$ is separating for $M'$. I proved ...
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23 views

A problem on left Fredholm Operator..

I was reading Fredholm Operators from the book "A course in Functinal Analysis " by J.B Conway. There I got stuck with the following problem. Let $A\in B(\mathcal H)$. Show that $A(\mathcal M)$ is ...
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Calculating operator (matrix) norms using eigenvalues?

A remark that went unproven in class. It was said that the operator norm of a real linear transformation (real matrix) is the square root of the abs value of the max eigenvalue of $A^T*A$ (or maybe ...
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33 views

Left Shift Operator Spectrum Q2

Consider $\ell^2(\mathbb{Z})$. Let $R: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be such that $R((a_n)) = (a_{n+1})$. I need to prove that, given $z \in \mathbb{C}$ with $|z| >1$, the two series ...
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26 views

Riemann Lebesgue Lemma for locally compact ableian groups

I'm looking for a reference (or proof here) of the generalized RL Lemma for LCAGs. One result is that if $G$ is a LCAG then $$\{ \hat{f} : f \in L^1(G)\} \subset C_0(\Gamma)$$ where $\Gamma$ is the ...
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62 views

Ranges of projection operators

Suppose that $X$ is a Banach space and $P$ and $Q$ be bounded linear projections on $X$ such that $PQ$ and $QP$ are compact. Does it follow that $PQ$ and $QP$ are finite-rank operators? My attempt: I ...
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57 views

Estimate spectral radius of operator product

In my research problem, I have to estimate the spectral radius of the following operator $\chi A$ where $\chi$ is a scalar function taking values 0 or 1 and $A$ is an operator. I can compute ...
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1answer
62 views

Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...
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53 views

Antiderivative as an integral operator from $L^2(0,2\pi)$ to $L^2(0,2\pi)$

I am starting to study Functional Analysis on Hilbert Spaces and I am studying the following operator: $$T:L^2(0,2\pi) \rightarrow L^2(0,2\pi) $$ where $$Tf:(0,2\pi) \rightarrow \mathbb{R} \\ ...
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Let $T$ be a definite integral operator on $(C[a,b])$. Find function $k_j$ such that $T^j (x)=\int_{a}^{t} k_j (s,t) x(s) \,ds$

This is the last part of chain of related questions: I was asked to prove that $$ T:C([a,b]) \to C([a,b]) $$ given by $$ Tx(t)=\int_{a}^{t} x(s) \, ds $$ is linear bounded operator on ...
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15 views

linear function, operator norm

Let be $\Phi:V\to W$ a linear function between the vector spaces $V$ and $W$ with the norms $\|\cdot\|_V$ and $\|\cdot\|_W$. Prove that $$\|\Phi\|_{\mathcal{L}(V,W)}=\kappa_{abs},$$ while ...
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55 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
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15 views

Number of solution of $ (\Delta - \lambda) f = \delta $

How they are solutions of the equation $$ (\Delta - \lambda) f (x) = \delta (x)$$ Where $\Delta$ is the Laplacian operator and $\delta(x) = \begin{cases} 0, \quad x\neq 0;\\ +\infty, \quad x= 0 ...
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44 views

Can $\text{ arg}$ be thought of as operator?

Forgive me if the question is to vague. The argument, denoted by $\text{arg}$, is a commonly used notation. I am specifically interested in the following use of $\text{arg}$: \begin{align} a=\text{ ...
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43 views

Topological characterization of the range of a bounded normal operator

Let $T$ be a bounded normal operator on a Hilbert space $H$. I want to prove the following statement: $\text{ran}(T)$ is closed if and only if 0 is not a limit point of $\sigma(T)$. I tried to use the ...
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4answers
59 views

Eigenvalues of an operator?

I have just started working with operators, ie objects that map functions to other functions, and I have heard people talking about the eigenavalues of an operator that can be obtained through ...
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1answer
25 views

Kaplansky density theorem

Let $H$ be a Hilbert space and $A$ a C*-subalgebra of $B(H)$, and $1_H\in A$. Show that the unitaries of $A$ are strongly dense in the unitaries of $\overline{A}^{sot}$. Suppose $U(A)$ be unitaries ...
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24 views

Pairs $(p,q)$ such that $id: l_p\to l_q$ is bounded [duplicate]

find all pairs $p,q\in [1,\infty)$ such that $id: l_p\to l_q$ is bounded. This just means I must find all $(p,q)$ such that $\|x\|_q \le C\|x\|_p$ for some $C$ dependent on $p$, $q$. I don't know ...
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Convergence of operator-exponential

Let $T: [0,\infty) \rightarrow L(X)$ define a $C_0$ semigroup on a Banach space $X$, then I want to show that $A_h:=\frac{T(h)-id}{h}$ are such that $e^{tA_h}(f) \rightarrow T(t)(f)$ pointwise. ...
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74 views

Frechet Derivatives of a nonlinear integral operator

The nonlinear integral operator $P:C[0,1]\to C[0,1]$ is defined as follow: $$P(f)(x)=1+kxf(x)\int_0^1\frac{f(s)}{x+s}ds$$ In order to obtain the Frechet derivative of the operator, I start with: ...
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155 views

Why are eigenfunctions of Laplace-Beltrami operators the minimizer of $\int_\mathcal{M}\| \nabla f(x)\|^2$?

Given a smooth $m$-dimensional manifold $\mathcal{M}$ embedded in $\Re^k$. Suppose we have a map $f : M \to \Re .$ Now, these are my questions: Specific question: i): Why does the $f$ that ...
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47 views

Show self-adjointness elementary

Is anybody aware of an elementary proof that $T^*T$ is self-adjoint where $T$ is closed and densely-defined? All proofs I found so far use the Friedrich's extension or other more sophisticated ...
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The domain of a root of a self-adjoint operator associated with an interpolation space

We now that $V$, $H$ are separable Hilbert spaces such that $V$ is dense in $H$ and $V\hookrightarrow H$ continuous, by representation theorem exists $A: D(A)\subset V\rightarrow H$ self adjoint e ...
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54 views

$D(T^*T)$ is a core for $T$.

Let $T$ be a closed densely-defined operator, then I want to show that $D(T^*T)$ is a core for $T$. This means the closure of $T|_{D_{T^*T}}$ is $T$ again. It is easy to notice that this is equivalent ...
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1answer
75 views

Contraction operator

In a proof of Picard's theorem using the contraction mapping theorem, we define an operator $T$ which is applied to a function $y$. I don't really see below how $Ty$ is any different from $y$ as the ...
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19 views

Derivative of an operator

I am trying to understand a few things about the following problem. I am given an operator $A(s)$, time dependent, positive definite and bounded (uniformly in time), boundedly invertible with compact ...
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308 views

Prove this map is not an open map

Let $K$ be the space of bounded, continuous real-valued functions $f$ from $(0, 1) \to \Bbb R$. Let $K$ have the supremum norm. Let $L: K \to K$ be defined by $L(f)(x) = x f(x)$. Show that $L$ is ...
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169 views

Prove that operator is surjective.

Take a sequence of bounded operators $S_n \in \mathcal{B}(X,X),$ where $X$ is a Banach space. Suppose that $S_n \rightarrow I,$ in the operator norm, for $n \to \infty.$ Then It´s easy to check that ...
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Holomorphic Functional Calculus for the Square Root

I'm working on a problem set, so I'm not looking for a solution, but just maybe a pointer on where I'm going wrong. I want to use the holomorphic functional calculus to determine the square root of ...
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Trace Class: Decomposition

This is only Q&A. Preview Trace class operators decompose. So proofs reduce to Hilbert-Schmidt! Problem Given a Hilbert Space $\mathcal{H}$. For the trace class: ...
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$\mathcal{L}^2$-norm of the Laplace transform

I have been considering the Laplace transform $$\mathcal{L}(f)(s)=\int_{0}^{\infty}{f(t)\, e^{-st}dt}$$ defined on $s\in\mathbb{R}^{+}$ as an linear operator from ...
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Approximating an element in the domain of an unbounded operator by a sequence in a dense subset of the domain.

Let $T$ be a closed unbounded (in my case also symmetric) operator on a Hilbert space $\mathcal{H}$ with dense domain $\mathcal{D}(T)$, and let $f\in \mathcal{D}(T)$. Suppose there is a dense ...
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26 views

Operator system of minimal dimension with one dimensional projections

Consider the matrix algebra $\mathbb{M}_n(\mathbb{C})$ with H-S inner producr ($\langle a, b\rangle =tr (a^*b)$). What is the minimal dimension of any operator system $\mathcal{A}$ in ...
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29 views

On suffienct condition on extending transpose of linear operator from dense subset to the closure.

Suppose we want to find the transpose of a linear operator on $L^{p}[a,b]$ to $L^{p}[a,b]$. If we slove the following equation $(Af,g)=(f,A^{*}g)$ for $g \in C[a,b]$ with the norm that makes its ...
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76 views

Is $B(H)$ sot separable

To prove that the unit ball of $B(H)$ is separable in strong operatior topology using the fact that $K(H)$ is separable and also is sot- dense in $B(H)$. I think we can conclude that $B(H)$ is also ...
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19 views

How do I solve this operator equation?

I am looking for a way to solve the following operator equation for an unknown operator $\hat{G}(\rho)$. $$ \frac{\textrm{d}}{\textrm{d} \rho}\hat{G}(\rho) = (-p)^l\frac{\textrm{d}}{\textrm{d} \rho} ...
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1answer
36 views

Determining the matrix of an abstract Linear Operator, with respect to a basis.

I've been struggling with the concept of finding the matrix of the operator, and need some help because I am preparing for an exam. I understand how to find the matrix of an operator/transformation ...
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1answer
36 views

projection in a factor von Neumann algebra.

We know that center of a factor von Neumann algebra $\mathcal{A} $ is trivial. Let $P_1$ be a projection in $\mathcal{A} $ such that $P_1\neq I,0$ . undoubtedly there exist another projection like ...
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If the bounded operators $X\to Y$ form a Banach space in the operator norm, is $Y$ necessarily Banach? [duplicate]

I have seen that if $Y$ is Banach, the set $B(X,Y)$ of bounded linear operators from $X$ to $Y$ is Banach in the operator norm. I was now wondering about the converse. Is it true? More precisely: ...
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Linear map from $L^1 \rightarrow L^{\infty}.$

I was wondering how I can show that any linear map $T: L^1(\Omega) \rightarrow L^{\infty}(\Omega)$ can be represented as an integral operator $$T(f)(x):=\int_{\Omega} K(x,y)f(y) dy.$$ Does anybody ...
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$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras.

Is it true that $A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$. I read in Murphy 2.1.11 that this is true if $B$ is ...
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29 views

Proof of the continuity method, guidance

Let $\mathcal{B}$ be a Banach space, and $V$ a normed linear space. $L_0,L_1:\mathcal{B}\to V$ are bounded linear operators. Assume $\exists c$ such that $L_t := (1-t)L_0 + tL_1$ satisfies: ...
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48 views

Self-adjoint and positive operator minimal polynomial on complex inner product spaces

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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23 views

Positive operator minimal polynomial [duplicate]

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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38 views

Dominated convergence theorem for spectral measure

Okay, I posed my question maybe a little bit to vague: What I have in mind is the following: Let $L$ be a generator of a semigroup $(P_t)_{t \ge 0}$ with $\langle x,Lx \rangle \le 0$ defined on ...
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35 views

Question about means on linear maps from vector space of bounded sequences to $\mathbb{R}$

The definitions I am working with: $B$ is the vector space of bounded sequences $a=(a_n), n\in\mathbb{Z}$ for which there exists $C>0$ such that $|a_n|\leq C, \forall n$ 2. Mean on $B$ is a ...
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Matrix monotone operators Intuition

can anyone explain by intuition that a matrix(operator) $A$ is monotone? I know for normal functions if a matrix is monotone this means intuitively i can think of it as increasing, but hard to ...
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Von neumann contains the range projections of all of its elements

The following is a theorem of Murphy's C*-algebra and operator theory: I think it can prove easier, while I'm not sure about my proof : Let $a\in A$ be positive. Consider $C^*(a)$, and let ...
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1answer
37 views

Spectrum of unbounded operators

I am currently a little bit confused. I am aware of a theorem that says that any closed and densely defined operator satisfies $\sigma(T^*)=\overline{\sigma(T)}.$ On the other hand, the operator ...