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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How to find spectrum of a convolution operator

Say $k$ be s.t. $\hat{k}$ is a bounded function on an LCA group $G$ and $Tf=f*k$. Then $T$ is bounded on $L^2(G)$. Is there anything I can say about $\sigma(T)$? (except the properties that follow ...
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35 views

The dual norm of a operator matrix norm

Let as look at matrices $B$ in $\mathbb{R}^{p\times q}$ together with the following operator norm: $$||B||_{op}:=\max_{\beta}\frac{|B\cdot \beta|_{p}}{|\beta|_{q}}.$$ Here $|\cdot|_{p}$ is any norm on ...
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32 views

Why an unbounded operator defined everywhere fails to be closed?

The Toeplitz theorem says : If a closed operator is defined everywhere, then it is continuous. So if a non continuous operator is defined everywhere, it is not closed. But why is it not closed? What ...
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18 views

Lumer-Phillips Theorem for non-contraction semigroups?

Let $H$ be a closed operator on a Hilbert space $\mathcal H$. The Lumer-Phillips theorem states that $H$ is a generator of a one-parameter contraction semigroup if and only if $\Re\langle ...
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1answer
30 views

Operator between Hilbert spaces, boundness, image and eigenvalues

I'm totally new in functional analysis and this is my first problem. Let's $H=L^2(-\pi,\pi)$ as Hilbert space with basis $u_n+iv_n$ where $$u_0 = ...
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75 views

Consider the Banach Space $C[0,1]$. Find decomposition of spectrum of the indefinite integral operator.

Cosider the Banach Space $C[0,1]$ of real-valued continuous function on $[0,1]$ with the supremum norm. and the linear operator $$A: x(t)\mapsto\int\limits_0^tx(s)ds$$ Find its eigenvalues, ...
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30 views

Existence of Unitary Map

I've been recently introduced to Unitary operators of a Hilbert space and I've been wondering the following. Existence of a unitary operator $T$ on a (possibly infinite) Hilbert space $H$ is simple ...
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55 views

Positive bounded operators

Let $A,B$ be positive self-adjoint bounded operators and $\lambda >0$ then I want to show that if $$A-B \ge 0 $$ that is $\langle x,(A-B)x \rangle \ge 0$ we have that the resolvents (whose ...
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28 views

The relation between Closed Operators (the graph is closed) and Closed Mappings (images of closed sets are closed)

Let $X$, and $Y$ be topological vector spaces and let $D$ be a dense vector subspace of $X$. An operator $T:D\to Y$ is called closed iff the graph of $T$, $\{(x,T(x))\in X\times Y|\,x\in D\}\subseteq ...
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28 views

Can we use a series of properties to determine integral operator $f \to \int_0^1 f d\mu $

Question: Suppose there exists an operator $I: C^{\infty}(0,1) \to \mathbb R$ satisfying the following properties: (1) $I (\chi_{(0,1)})=1$ ; (2) $I(kf)=kI(f)$, where $k\in \mathbb R$ and $f\in ...
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32 views

prove that all pure states in a commutative C* algebra are multiplicative linear functionals

I am trying to prove this , but can not see it clearly. it was given as some sort of converse of the fact that all multiplicative linear functionals are pure states
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26 views

Find basis for exact matrix form of linear operator

$A:\cal{P}_1 \to \cal{P}_1$ is a linear operator defined with $$ A(p)(t):=(3t+1)p'(t)+2p(t). $$ I'm trying to find a basis $e$ of $\cal{P}_1$ in which the operator $A$ has the matrix form $$ A= ...
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51 views

Find the eigenvalues of the operator T.

I have the following problem, "Suppose that $X=\ell^1$ and define the operator $T\in B(X)$ as follows: $$Tx=\left(\frac12x_2,\frac13x_3,\frac14x_4,...\right)\,,\textit{where,}\,\,\, ...
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1answer
28 views

Inverse continuity of an operator

Let $X$ be a Banach space (it is in fact an $L^p$ space) and let $T:X \to X$ be a linear continuous operator (which is not injective and not surjective). I am trying to figure out if the following is ...
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43 views

Fundamental solution of a shifted operator

what is the fundamental solution of the shifted operator $ \Delta + \lambda^2 $, i.e, what the function $f$ satisfying the following equation $$ (\Delta + \lambda^2 )f(x) = \delta(x),$$ where $ \Delta ...
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185 views

idempotents in a subalgebra of $B(H)$.

Let $\mathcal{A}$ be a sub-algebra of $B(H)$ such that $\mathcal{A}$ generated by all its idempotents and $\mathcal{A}$ is closed under weak operator topology. Suppose that there exist idempotents ...
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18 views

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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26 views

$\omega$ 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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19 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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39 views

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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10 views

Operatorial norm for matrixes and maximal element

I have two matrices, say $A$ and $B$ have operatorial norm equal to 1 and they are in $M_{n×m}(\mathbb(R))$. Let $u,v\in\mathbb(R)$ be the vectors such that $\lvert Au\rvert=1$ and $\lvert ...
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1answer
25 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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20 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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61 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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1answer
47 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
50 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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1answer
49 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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11 views

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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1answer
14 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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50 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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11 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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30 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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1answer
40 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
51 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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23 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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22 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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48 views

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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1answer
71 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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1answer
152 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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43 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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50 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
56 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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18 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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1answer
298 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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127 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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1answer
36 views

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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1answer
25 views

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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89 views

$\mathcal{L}^2$-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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44 views

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 ...