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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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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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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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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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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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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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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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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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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74 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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16 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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22 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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31 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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47 views

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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32 views

$A^+ \subset B^+$ 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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25 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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43 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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22 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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35 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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34 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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26 views

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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30 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 ...
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26 views

compact and normal operator is diagonalizable

In the following theorm, I do not know why $K^\perp = 0$ I just accept that $x = 0$ on $K^\perp$. For instance, $x = h\otimes h$ for $h\in H_{\|.\|=1}$ is a compact operator. Extend $\{h\}$ to a ...
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Power series coming from linear function in $\ell_3^*$

The Problem Suppose that $f\in \ell_3^*$. Show that the series $\sum_{n=1}^\infty f(e_n)^3$ converges. Discussion This problem is on a practice exam I have for a linear analysis course. My first ...
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Trace Class: Counterexample

This is a real question! Given a Hilbert space $\mathcal{H}$. Denote trace class by:* $$\mathcal{B}_\textrm{Tr}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):\operatorname{Tr}|A|<\infty\}$$ Then ...
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How can I get eigenvalues of infinite dimensional linear operator?

What I want to prove is that for infinite dimensional vector space, $0$ is the only eigenvalue doesn't imply $T$ is nilpotent. But I am not sure how to find eigenvalues of infinite dimensional linear ...
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27 views

For what operators $A$ on a Hilbert space is the identity operator in the closure of the similarity orbit of $A$?

For a bounded linear operator $A$ on a separable Hilbert space, the similarity orbit of $A$ is the set $S(A)=\{WAW^{-1}: W \text{ is invertible}\}$. I am wondering that if the identity operator $I$ is ...
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20 views

Finite measure operator norm

Let $T^n: L^1(\mu) \rightarrow L^{\infty}(\mu)$ be a bounded operator for any $n$ and $\mu$ a probability measure. Is it then true that $||T^2||_{1 \rightarrow \infty} \le ||T||_{1 \rightarrow 2} ...
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39 views

Resolvent set/operator

Just a question here. Why do we study Resolvent operators and resolvent sets? Will there be any motivation or intuition behind this?
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Operator norm of positive operator.

I'm studying Reed and Simon's "Methods of Modern Mathematical Physics" Vol. 1 (http://www.math.bme.hu/~balint/oktatas/fun/notes/Reed_Simon_Vol1.pdf). In the proof of the square root lemma (p.196) they ...
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26 views

Definition of “Extension” of Bounded Linear Transformation

I have been given the problem of proving the B.L.T. Theorem for my homework which states, Every bounded linear transformation $\mathsf{T}$ from a normed vector space X to a complete, normed vector ...
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27 views

A restriction of a symmetric operator such that the range of (operator)+i is the same

I have this problem and I really can't see how to do it. Suppose that $C$ is a symmetric operator, $A\subset C$ and that $\operatorname{Ran}(C+i)=\operatorname{Ran}(A+i)$. Prove that $C=A$. ...
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20 views

Why is $\sqrt{T^*T}$ self-adjoint?

Let $T$ be a bounded linear operator over some Hilbert space $H$. Since $T^*T$ is a positive operator, it has a square root. Let $R=\sqrt{T^*T}$. Prove that $\forall u\in H, ||Ru||=||Tu||$. ...
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Existence of operator with certain properties on a Banach space

I ran across this question, and was a little puzzled by it. I neither know how to solve it, nor its meaning: Let $X$ be a Banach space, and let $A,B$ be bounded linear operators on $X$ such that $A$ ...
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20 views

Given an operator $ * $ and it's inverse $ \setminus $ when does $ x \setminus y = x * \left( 0 \setminus y \right) $?

Given a groupoid $ \left( M, * \right) $ with an neutral element $0$ and $\setminus$ being an inverse operator of $*$, what are the groupoid's properties for this predicate to be true? $$ \forall x, ...
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118 views

Trace Class: Relativeness

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint: $$H\in\mathcal{B}(\mathcal{H}):\quad H=H^*$$ Denote trace class: ...
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Spectrum of a closed operator

Could someone please explain this fact: if $A$ is a closed operator and $A^{-1}$ is a compact operator, then spectrum of $A$ consist only of eigenvalues? I forgot to mention that operator $A^{-1}$ is ...
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Conjugation: Boundedness [closed]

Given a Banach space $E$. Then for conjugations: $$C:E\to E:\quad C^2=1\implies\|C\|=1$$ How can I check this?
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69 views

What does $\operatorname{supp}(A)$ mean?

I'm looking at a paper (specifically this one). In the paper, we have a positive operator $A$, and the operator $\operatorname{supp}(A)$ is supposed to be a projection operator. Does anybody know ...
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What is the representation of the Grushin operator $G_\alpha$ in spherical coordinates $(r, \theta)$?

We known that the Laplace operator in spherical coordinates $(r, \theta)$ where $r = |x|$ and $\theta =\frac{x}{|x|} \in S^{N-1}$ is $$\Delta u = u''_{rr} + u'_r + r^{-2}\Delta_{S^{N-1}}$$ here ...
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Is the following differential operator closed (closabe)?

Let $L$ be the following differential operator. $L: C^2(\Bbb{R}^2_+)\to C^0(\Bbb{R}^2_+) $ $$Lf = \partial_x f(x,y) (y-x) + \partial_y f (x,y)(x-y) + \frac{1}{2} \bigg( \partial_{xx} f(x,y)x + ...
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26 views

transform between unitary operators

If I have a unitary operator $\exp(i\phi X)$, where $X$ is hermitian and $\phi\in\mathbb{R}$, is there a known way of finding an operator $\hat Y$ such that $$\exp(i\phi \hat X)=\exp(if(\phi) \hat ...
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Integration of $A$-valued functions (Functional Analysis)

Premise 1: my source is the Rudin - Functional Analysis. Premise 2: i'm not a mathmo so forgive for the mistakes A couple of question on the subject... An example of Banach Algebra is the set of ...
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C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let ...